{"id":218,"date":"2019-04-09T01:14:20","date_gmt":"2019-04-09T05:14:20","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&#038;p=218"},"modified":"2019-04-12T19:09:27","modified_gmt":"2019-04-12T23:09:27","slug":"4-1-the-hydrogen-atom","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/4-1-the-hydrogen-atom\/","title":{"raw":"4.1 The Hydrogen Atom","rendered":"4.1 The Hydrogen Atom"},"content":{"raw":"<div data-type=\"abstract\" id=\"45781\" class=\"ui-has-child-title\"><header>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Describe the hydrogen atom in terms of wave function, probability density, total energy, and orbital angular momentum<\/li>\r\n \t<li>Identify the physical significance of each of the quantum numbers (<span class=\"MathJax_MathML\" id=\"MathJax-Element-577-Frame\"><span class=\"MathJax_MathContainer\"><span>n,l,m<\/span><\/span><\/span>) of the hydrogen atom<\/li>\r\n \t<li>Distinguish between the Bohr and Schr\u00f6dinger models of the atom<\/li>\r\n \t<li>Use quantum numbers to calculate important information about the hydrogen atom<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The hydrogen atom is the simplest atom in nature and, therefore, a good starting point to study atoms and atomic structure. The hydrogen atom consists of a single negatively charged electron that moves about a positively charged proton (<\/span>Figure 4.2<span style=\"font-size: 14pt\">). In Bohr\u2019s model, the electron is pulled around the proton in a perfectly circular orbit by an attractive Coulomb force. The proton is approximately 1800 times more massive than the electron, so the proton moves very little in response to the force on the proton by the electron. (This is analogous to the Earth-Sun system, where the Sun moves very little in response to the force exerted on it by Earth.) An explanation of this effect using Newton\u2019s laws is given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:fdd4b413-6910-44d5-801d-0f4223bc7a31@5\" data-page=\"48\" style=\"font-size: 14pt\">Photons and Matter Waves<\/a><span style=\"font-size: 14pt\">.<\/span>\r\n\r\n<\/header><\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_HydAtom\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"429\"]<img alt=\"The Bohr model of the hydrogen atom has the proton, charge q = plus e, at the center and the electron, charge q = minus e, in a circular orbit centered on the proton.\" data-media-type=\"image\/jpeg\" id=\"57666\" src=\"https:\/\/cnx.org\/resources\/a5587a5541704e45cfdf7c6152d84a08d4a569e2\" width=\"429\" height=\"275\" \/> Figure 4.2 A representation of the Bohr model of the hydrogen atom.[\/caption]<\/figure>\r\n<div class=\"os-caption-container\">With the assumption of a fixed proton, we focus on the motion of the electron.<\/div>\r\n<\/div>\r\nIn the electric field of the proton, the potential energy of the electron is\r\n<div data-type=\"equation\" id=\"fs-id1170903082088\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-578-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-578-Frame\"><span class=\"MathJax_MathContainer\"><span>U(r)=\u2212ke2r,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.1]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-579-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">k=1\/4\u03c0\u03b50<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the distance between the electron and the proton. As we saw earlier, the force on an object is equal to the negative of the gradient (or slope) of the potential energy function. For the special case of a hydrogen atom, the force between the electron and proton is an attractive Coulomb force.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902719456\">Notice that the potential energy function<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>) does not vary in time. As a result,<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term329\">Schr\u00f6dinger\u2019s equation<\/span><span>\u00a0<\/span>of the hydrogen atom reduces to two simpler equations: one that depends only on space (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>) and another that depends only on time (<em data-effect=\"italics\">t<\/em>). (The separation of a wave function into space- and time-dependent parts for time-independent potential energy functions is discussed in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:018ccaf5-d217-467a-b03a-d2ea9d46f6ee\" data-page=\"56\">Quantum Mechanics<\/a>.) We are most interested in the space-dependent equation:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170903052286\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-580-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-580-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22me(\u22022\u03c8\u2202x2+\u22022\u03c8\u2202y2+\u22022\u03c8\u2202z2)\u2212ke2r\u03c8=E\u03c8,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.2]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-581-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8=\u03c8(x,y,z)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the three-dimensional wave function of the electron,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-582-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">me<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the mass of the electron, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the total energy of the electron. Recall that the total wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-583-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,y,z,t),<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the product of the space-dependent wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-584-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8=\u03c8(x,y,z)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the time-dependent wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-585-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c6=\u03c6(t)<\/span><\/span><span style=\"font-size: 14pt\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902893842\">In addition to being time-independent,<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>) is also spherically symmetrical. This suggests that we may solve Schr\u00f6dinger\u2019s equation more easily if we express it in terms of the spherical coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-586-Frame\"><span class=\"MathJax_MathContainer\"><span>(r,\u03b8,\u03d5)<\/span><\/span><\/span><span>\u00a0<\/span>instead of rectangular coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-587-Frame\"><span class=\"MathJax_MathContainer\"><span>(x,y,z)<\/span><\/span><\/span>. A spherical coordinate system is shown in<span>\u00a0<\/span>Figure 4.3. In spherical coordinates, the variable<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>is the radial coordinate,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-588-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>is the polar angle (relative to the vertical<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis), and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-589-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>is the azimuthal angle (relative to the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>-axis). The relationship between spherical and rectangular coordinates is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-590-Frame\"><span class=\"MathJax_MathContainer\"><span>x=rsin\u03b8cos\u03d5,y=rsin\u03b8sin\u03d5,z=rcos\u03b8.<\/span><\/span><\/span><\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_SphCoord\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"418\"]<img alt=\"An x y z coordinate system is shown, along with a point P and the vector r from the origin to P. In this figure, the point P has positive x, y, and z coordinates. The vector r is inclined by an angle theta from the positive z axis. Its projection on the x y plane makes an angle theta from the positive x axis toward the positive y axis.\" data-media-type=\"image\/jpeg\" id=\"80779\" src=\"https:\/\/cnx.org\/resources\/233c5feaf72800977965229aeb9225feadaab04b\" width=\"418\" height=\"379\" \/> Figure 4.3 The relationship between the spherical and rectangular coordinate systems.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170903095566\">The factor<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-591-Frame\"><span class=\"MathJax_MathContainer\"><span>rsin\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>is the magnitude of a vector formed by the projection of the polar vector onto the<span>\u00a0<\/span><em data-effect=\"italics\">xy<\/em>-plane. Also, the coordinates of<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em><span>\u00a0<\/span>are obtained by projecting this vector onto the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>- and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>-axes, respectively. The inverse transformation gives<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903139998\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-592-Frame\"><span class=\"MathJax_MathContainer\"><span>r=x2+y2+z2,\u03b8=cos\u22121(zr),\u03d5=cos\u22121(xx2+y2).<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903044398\">Schr\u00f6dinger\u2019s wave equation for the hydrogen atom in spherical coordinates is discussed in more advanced courses in modern physics, so we do not consider it in detail here. However, due to the spherical symmetry of<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>), this equation reduces to three simpler equations: one for each of the three coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-593-Frame\"><span class=\"MathJax_MathContainer\"><span>(r,\u03b8,and\u03d5).<\/span><\/span><\/span><span>\u00a0<\/span>Solutions to the time-independent wave function are written as a product of three functions:<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902913459\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-594-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(r,\u03b8,\u03d5)=R(r)\u0398(\u03b8)\u03a6(\u03d5),<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902898644\">where<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>is the radial function dependent on the radial coordinate<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>only;<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-595-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0398<\/span><\/span><\/span><span>\u00a0<\/span>is the polar function dependent on the polar coordinate<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-596-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>only; and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-597-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a6<\/span><\/span><\/span><span>\u00a0<\/span>is the phi function of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-598-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>only. Valid solutions to Schr\u00f6dinger\u2019s equation<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-599-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(r,\u03b8,\u03d5)<\/span><\/span><\/span><span>\u00a0<\/span>are labeled by the quantum numbers<span>\u00a0<\/span><em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>.<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901533514\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-600-Frame\"><span class=\"MathJax_MathContainer\"><span>n:principal quantum number l:angular momentum quantum number m:angular momentum projection quantum number<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902666863\">(The reasons for these names will be explained in the next section.) The radial function<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">n<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>; the polar function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-601-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0398<\/span><\/span><\/span><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>; and the phi function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-602-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a6<\/span><\/span><\/span><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>. The dependence of each function on quantum numbers is indicated with subscripts:<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901741972\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-603-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm(r,\u03b8,\u03d5)=Rnl(r)\u0398lm(\u03b8)\u03a6m(\u03d5).<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902925786\">Not all sets of quantum numbers (<em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>) are possible. For example, the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can never be greater or equal to the principal quantum number<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-604-Frame\"><span class=\"MathJax_MathContainer\"><span>n(l&lt;n)<\/span><\/span><\/span>. Specifically, we have<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902660979\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-605-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,2,3,\u2026l=0,1,2,\u2026,(n\u22121)m=\u2212l,(\u2212l+1),\u2026,0,\u2026,(+l\u22121),+l<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901578315\">Notice that for the ground state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-606-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-607-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-608-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0<\/span><\/span><\/span>. In other words, there is only one quantum state with the wave function for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-609-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span>, and it is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-610-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100<\/span><\/span><\/span>. However, for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-611-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>, we have<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902941717\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-612-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0,m=0l=1,m=\u22121,0,1.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901800555\">Therefore, the allowed states for the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-613-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><span>\u00a0<\/span>state are<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-614-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-615-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c821\u22121,\u03c8210<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-616-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8211<\/span><\/span><\/span>. Example wave functions for the hydrogen atom are given in<span>\u00a0<\/span>Table 4.1. Note that some of these expressions contain the letter<span>\u00a0<\/span><em data-effect=\"italics\">i<\/em>, which represents<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-617-Frame\"><span class=\"MathJax_MathContainer\"><span>\u22121<\/span><\/span><\/span>. When probabilities are calculated, these complex numbers do not appear in the final answer.<\/p>\r\n\r\n<div class=\"os-table\">\r\n<table id=\"fs-id1170901599432\" summary=\"Table 8.1 Wave Functions of the Hydrogen Atom \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-618-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,l=0,ml=0<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-619-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100=1\u03c01a03\/2e\u2212r\/a0<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-620-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=0,ml=0<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-621-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200=142\u03c01a03\/2(2\u2212ra0)e\u2212r\/2a0<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-622-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=\u22121<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-623-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c821\u22121=18\u03c01a03\/2ra0e\u2212r\/2a0sin\u03b8e\u2212i\u03d5<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-624-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=0<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-625-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8210=142\u03c01a03\/2ra0e\u2212r\/2a0cos\u03b8<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-626-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=1<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-627-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8211=18\u03c01a03\/2ra0e\u2212r\/2a0sin\u03b8ei\u03d5<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Wave Functions of the Hydrogen Atom<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\r\n<\/div>\r\n<section id=\"fs-id1170902884504\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Physical Significance of the Quantum Numbers<\/h3>\r\n<p id=\"fs-id1170902884509\">Each of the three quantum numbers of the hydrogen atom (<em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>) is associated with a different physical quantity. The<span>\u00a0<\/span><span data-type=\"term\" id=\"term330\">principal quantum number<\/span><span>\u00a0<\/span><em data-effect=\"italics\">n<\/em><span>\u00a0<\/span>is associated with the total energy of the electron,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-628-Frame\"><span class=\"MathJax_MathContainer\"><span>En<\/span><\/span><\/span>. According to Schr\u00f6dinger\u2019s equation:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902869741\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-629-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-629-Frame\"><span class=\"MathJax_MathContainer\"><span>En=\u2212(mek2e422)(1n2)=\u2212E0(1n2),<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.3]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-630-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E0=\u221213.6eV.<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Notice that this expression is identical to that of Bohr\u2019s model. As in the Bohr model, the electron in a particular state of energy does not radiate.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1170902858591\" class=\"ui-has-child-title\"><header><\/header><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\"><\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170902858593\"><span data-type=\"title\"><strong>How Many Possible States?<\/strong><\/span><\/p>\r\nFor the hydrogen atom, how many possible quantum states correspond to the principal number<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-631-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>? What are the energies of these states?\r\n\r\n<span data-type=\"title\"><strong>Strategy<\/strong><\/span>\r\n\r\nFor a hydrogen atom of a given energy, the number of allowed states depends on its orbital angular momentum. We can count these states for each value of the principal quantum number,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-632-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,2,3.<\/span><\/span><\/span><span>\u00a0<\/span>However, the total energy depends on the principal quantum number only, which means that we can use<span>\u00a0<\/span>Equation 4.3<span>\u00a0<\/span>and the number of states counted.\r\n\r\n<span data-type=\"title\"><strong>Solution<\/strong><\/span>\r\n\r\nIf<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-633-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>, the allowed values of<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>are 0, 1, and 2. If<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-634-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-635-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0<\/span><\/span><\/span><span>\u00a0<\/span>(1 state). If<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-636-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-637-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u22121,0,+1<\/span><\/span><\/span><span>\u00a0<\/span>(3 states); and if<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-638-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-639-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u22122,\u22121,0,+1,+2<\/span><\/span><\/span><span>\u00a0<\/span>(5 states). In total, there are<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-640-Frame\"><span class=\"MathJax_MathContainer\"><span>1+3+5=9<\/span><\/span><\/span><span>\u00a0<\/span>allowed states. Because the total energy depends only on the principal quantum number,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-641-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>, the energy of each of these states is\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901632559\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-642-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>En3=\u2212E0(1n2)=\u221213.6eV9=\u22121.51eV.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">An electron in a hydrogen atom can occupy many different angular momentum states with the very same energy. As the orbital angular momentum increases, the number of the allowed states with the same energy increases.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<span style=\"font-size: 14pt\">The<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term331\" style=\"font-size: 14pt\">angular momentum orbital quantum number<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">l<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is associated with the orbital angular momentum of the electron in a hydrogen atom. Quantum theory tells us that when the hydrogen atom is in the state<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-643-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8nlm<\/span><\/span><span style=\"font-size: 14pt\">, the magnitude of its orbital angular momentum is<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-644-Frame\"><span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210f,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.4]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span>\r\n\r\n<\/section><\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901842631\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-645-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0,1,2,\u2026,(n\u22121).<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903077401\">This result is slightly different from that found with Bohr\u2019s theory, which quantizes angular momentum according to the rule<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-646-Frame\"><span class=\"MathJax_MathContainer\"><span>L=n,where n=1,2,3,....<\/span><\/span><\/span><\/p>\r\n<p id=\"fs-id1170901596724\">Quantum states with different values of orbital angular momentum are distinguished using<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term332\">spectroscopic notation<\/span><span>\u00a0<\/span>(Table 4.2). The designations<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">f<\/em><span>\u00a0<\/span>result from early historical attempts to classify atomic spectral lines. (The letters stand for sharp, principal, diffuse, and fundamental, respectively.) After<span>\u00a0<\/span><em data-effect=\"italics\">f<\/em>, the letters continue alphabetically.<\/p>\r\n<p id=\"fs-id1170902899390\">The<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term333\">ground state<\/span><span>\u00a0<\/span>of hydrogen is designated as the 1<em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state, where \u201c1\u201d indicates the energy level<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-647-Frame\"><span class=\"MathJax_MathContainer\"><span>(n=1)<\/span><\/span><\/span><span>\u00a0<\/span>and \u201c<em data-effect=\"italics\">s<\/em>\u201d indicates the orbital angular momentum state (<span class=\"MathJax_MathML\" id=\"MathJax-Element-648-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>). When<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-649-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can be either 0 or 1. The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-650-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-651-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><span>\u00a0<\/span>state is designated \u201c2<em data-effect=\"italics\">s<\/em>.\u201d The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-652-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-653-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>state is designated \u201c2<em data-effect=\"italics\">p<\/em>.\u201d When<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-654-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can be 0, 1, or 2, and the states are 3<em data-effect=\"italics\">s<\/em>, 3<em data-effect=\"italics\">p<\/em>, and 3<em data-effect=\"italics\">d<\/em>, respectively. Notation for other quantum states is given in<span>\u00a0<\/span>Table 4.3.<\/p>\r\n<p id=\"fs-id1170902891111\">The<span>\u00a0<\/span><span data-type=\"term\" id=\"term334\">angular momentum projection quantum number<\/span><span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>is associated with the azimuthal angle<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-655-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>(see<span>\u00a0<\/span>Figure 4.3) and is related to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of orbital angular momentum of an electron in a hydrogen atom. This component is given by<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-656-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz=m\u210f,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.5]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901867971\">where<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901621612\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-657-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u2212l,\u2212l+1,\u2026,0,\u2026,+l\u22121,l.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902849835\">The<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of angular momentum is related to the magnitude of angular momentum by<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170903125406\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-658-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-658-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz=Lcos\u03b8,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.6]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-659-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the angle between the angular momentum vector and the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis. Note that the direction of the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis is determined by experiment\u2014that is, along any direction, the experimenter decides to measure the angular momentum. For example, the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-direction might correspond to the direction of an external magnetic field. The relationship between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-660-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">Lz and L<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.4.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_ZcompAng\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"387\"]<img alt=\"An x y z coordinate system is shown. The vector L is at an angle theta to the positive z axis and has positive z component L sub z equal to m times h bar. The x and y components are positive but not specified.\" data-media-type=\"image\/jpeg\" id=\"63328\" src=\"https:\/\/cnx.org\/resources\/688e44bcedab2fdb4d80de7eeb83c8e20c1b9232\" width=\"387\" height=\"355\" \/> Figure 4.4 The z-component of angular momentum is quantized with its own quantum number m.[\/caption]<\/figure>\r\n<\/div>\r\n<div class=\"os-table\">\r\n<table id=\"fs-id1170902779373\" summary=\"Table 8.2 Spectroscopic Notation and Orbital Angular Momentum \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Orbital Quantum Number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><\/th>\r\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Angular Momentum<\/th>\r\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">State<\/th>\r\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Spectroscopic Name<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">0<\/td>\r\n<td data-align=\"left\" data-valign=\"top\">0<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">Sharp<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">1<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-661-Frame\"><span class=\"MathJax_MathContainer\"><span>2h<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">Principal<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">2<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-662-Frame\"><span class=\"MathJax_MathContainer\"><span>6h<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">d<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">Diffuse<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">3<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-663-Frame\"><span class=\"MathJax_MathContainer\"><span>12h<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">f<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">Fundamental<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">4<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-664-Frame\"><span class=\"MathJax_MathContainer\"><span>20h<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">g<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\">5<\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-665-Frame\"><span class=\"MathJax_MathContainer\"><span>30h<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">h<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Spectroscopic Notation and Orbital Angular Momentum<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\r\n<\/div>\r\n<div class=\"os-table\">\r\n<table id=\"fs-id1170903078180\" summary=\"Table 8.3 Spectroscopic Description of Quantum States \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-666-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-667-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-668-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-669-Frame\"><span class=\"MathJax_MathContainer\"><span>l=3<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-670-Frame\"><span class=\"MathJax_MathContainer\"><span>l=4<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-671-Frame\"><span class=\"MathJax_MathContainer\"><span>l=5<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-672-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">1<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-673-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">2<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">2<em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-674-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">d<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-675-Frame\"><span class=\"MathJax_MathContainer\"><span>n=4<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">d<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">f<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-676-Frame\"><span class=\"MathJax_MathContainer\"><span>n=5<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">d<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">f<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">g<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-677-Frame\"><span class=\"MathJax_MathContainer\"><span>n=6<\/span><\/span><\/span><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">s<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">p<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">d<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">f<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">g<\/em><\/td>\r\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">h<\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Spectroscopic Description of Quantum States<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901531308\">The quantization of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-678-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz<\/span><\/span><\/span><span>\u00a0<\/span>is equivalent to the quantization of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-679-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>. Substituting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-680-Frame\"><span class=\"MathJax_MathContainer\"><span>l(l+1)\u210f<\/span><\/span><\/span><span>\u00a0<\/span>for<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-681-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz<\/span><\/span><\/span><span>\u00a0<\/span>into this equation, we find<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902680809\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-682-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-682-Frame\"><span class=\"MathJax_MathContainer\"><span>m\u210f=l(l+1)\u210fcos\u03b8.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.7]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Thus, the angle<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-683-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is quantized with the particular values<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901754937\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-684-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-684-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8=cos\u22121(ml(l+1)).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.8]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Notice that both the polar angle (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-685-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">) and the projection of the angular momentum vector onto an arbitrary<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-686-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"font-size: 14pt\">) are quantized.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901648045\">The quantization of the polar angle for the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-687-Frame\"><span class=\"MathJax_MathContainer\"><span>l=3<\/span><\/span><\/span><span>\u00a0<\/span>state is shown in<span>\u00a0<\/span>Figure 4.5. The orbital angular momentum vector lies somewhere on the surface of a cone with an opening angle<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-688-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>relative to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis (unless<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-689-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0,<\/span><\/span><\/span><span>\u00a0<\/span>in which case<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-690-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8=90\u00b0<\/span><\/span><\/span><span>\u00a0<\/span>and the vector points are perpendicular to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis).<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_QuantTheta\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"349\"]<img alt=\"Seven vector, all of the same length L, are drawn at 7 different angles to the z axis. The z components of the vectors are indicated both by horizontal lines from the tip of the vector to the z axis and by labels on the z axis. For four of the vectors, the angle between the z axis and the vector is also labeled. The z component values are 3 h bar at angle theta sub three, 2 h bar at angle theta sub two, h bar at angle theta sub one, zero at angle theta sub zero, minus h bar, minus 2 h bar, and minus 3 h bar.\" data-media-type=\"image\/jpeg\" id=\"77046\" src=\"https:\/\/cnx.org\/resources\/0702c75c508115325b2f61409dae642d3dfca9df\" width=\"349\" height=\"496\" \/> Figure 4.5 The quantization of orbital angular momentum. Each vector lies on the surface of a cone with axis along the z-axis.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170901577555\">A detailed study of angular momentum reveals that we cannot know all three components simultaneously. In the previous section, the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of orbital angular momentum has definite values that depend on the quantum number<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>. This implies that we cannot know both<span>\u00a0<\/span><em data-effect=\"italics\">x-<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>-components of angular momentum,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-691-Frame\"><span class=\"MathJax_MathContainer\"><span>Lx<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-692-Frame\"><span class=\"MathJax_MathContainer\"><span>Ly<\/span><\/span><\/span>, with certainty. As a result, the precise direction of the orbital angular momentum vector is unknown.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1170902705884\" class=\"ui-has-child-title\"><header><\/header><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.2<\/span><span class=\"os-divider\"><\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170902705886\"><span data-type=\"title\"><strong>What Are the Allowed Directions?<\/strong><\/span><\/p>\r\nCalculate the angles that the angular momentum vector<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-693-Frame\"><span class=\"MathJax_MathContainer\"><span>L\u2192<\/span><\/span><\/span><span>\u00a0<\/span>can make with the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-694-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span>, as shown in<span>\u00a0<\/span>Figure 4.6.\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_AngMoment\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"431\"]<img alt=\"The image shows three possible values of a component of a given angular momentum along z-axis. The upper circular orbit is shown for m sub t = 1 at a distance L sub z above the origin. The vector L makes an angle of theta one with the z axis. The radius of the orbit is the component of L perpendicular to the z axis. The middle circular orbit is shown for m sub t = 0. It is in the x y plane. The vector L makes an angle of theta two of 90 degrees with the z axis. The radius of the orbit is L. The lower circular orbit is shown for m sub t = -1 at a distance L sub z below the origin. The vector L makes an angle of theta three with the z axis. The radius of the orbit is the component of L perpendicular to the z axis.\" data-media-type=\"image\/jpeg\" id=\"64105\" src=\"https:\/\/cnx.org\/resources\/8eab941cffacd2c77a675ec48b93cf415e951c33\" width=\"431\" height=\"571\" \/> Figure 4.6 The component of a given angular momentum along the z-axis (defined by the direction of a magnetic field) can have only certain values. These are shown here for l=1, for which m=\u22121,0,and+1. The direction of L\u2192 is quantized in the sense that it can have only certain angles relative to the z-axis.[\/caption]<\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\">\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span><\/span><\/span><\/span>&nbsp;\r\n\r\n<span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">The vectors<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-698-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-699-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192z<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">(in the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction) form a right triangle, where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-700-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the hypotenuse and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-701-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192z<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the adjacent side. The ratio of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-702-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to |<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-703-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">| is the cosine of the angle of interest. The magnitudes<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-704-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L=|L\u2192|<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-705-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">are given by<\/span>\r\n<\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902866146\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-706-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210fandLz=m\u210f.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">We are given<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-707-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">ml<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can be<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-708-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">+1,0,or\u22121.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">Thus,<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">has the value given by<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902861390\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-709-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210f=2\u210f.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The quantity<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-710-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can have three values, given by<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-711-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz=ml\u210f<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902897710\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-712-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>Lz=ml\u210f={\u210f,ml=+10,ml=0\u2212\u210f,ml=\u22121<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">As you can see in<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span>Figure 4.6<span style=\"text-indent: 1em;font-size: 1rem\">,<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-713-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b8=Lz\/L,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">so for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-714-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">m=+1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, we have<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902923917\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-715-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>cos\u03b81=LZL=\u210f2\u210f=12=0.707.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Thus,<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902652024\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-716-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03b81=cos\u221210.707=45.0\u00b0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Similarly, for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-717-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">m=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, we find<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-718-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b82=0;<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">this gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901489064\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-719-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03b82=cos\u221210=90.0\u00b0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Then for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-720-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml=\u22121<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902765413\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-721-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>cos\u03b83=LZL=\u2212\u210f2\u210f=\u221212=\u22120.707,<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">so that<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902922844\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-722-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03b83=cos\u22121(\u22120.707)=135.0\u00b0.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The angles are consistent with the figure. Only the angle relative to the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis is quantized.<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can point in any direction as long as it makes the proper angle with the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis. Thus, the angular momentum vectors lie on cones, as illustrated. To see how the correspondence principle holds here, consider that the smallest angle (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-723-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03b81<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">in the example) is for the maximum value of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-724-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">namely<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-725-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml=l.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">For that smallest angle,<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901642419\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-726-Frame\"><span class=\"MathJax_MathContainer\"><span>cos\u03b8=LzL=ll(l+1),\r\n\r\n<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">which approaches 1 as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">becomes very large. If<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-727-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b8=1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, then<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-728-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03b8=0\u00ba<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. Furthermore, for large<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">, there are many values of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-729-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so that all angles become possible as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">gets very large.<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1170901751543\" data-depth=\"1\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a04<\/span><\/span><span class=\"os-number\">.1<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header><span style=\"font-size: 1rem\">Can the magnitude of<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-730-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">ever be equal to<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">?<\/span><\/header><\/div>\r\n<\/div>\r\n<h3 data-type=\"title\">Using the Wave Function to Make Predictions<\/h3>\r\n<p id=\"fs-id1170901751548\">As we saw earlier, we can use quantum mechanics to make predictions about physical events by the use of probability statements. It is therefore proper to state, \u201cAn electron is located within this volume with this probability at this time,\u201d but not, \u201cAn electron is located at the position (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>) at this time.\u201d To determine the probability of finding an electron in a hydrogen atom in a particular region of space, it is necessary to integrate the probability density<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-731-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8nlm|2<\/span><\/span><\/span><span>\u00a0<\/span>over that region:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902682281\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-732-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-732-Frame\"><span class=\"MathJax_MathContainer\"><span>Probability=\u222bvolume|\u03c8nlm|2dV,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.9]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dV<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is an infinitesimal volume element. If this integral is computed for all space, the result is 1, because the probability of the particle to be located<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">somewhere<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is 100% (the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term335\" style=\"font-size: 14pt\">normalization<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">condition). In a more advanced course on modern physics, you will find that<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-733-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8nlm|2=\u03c8nlm*\u03c8nlm,<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-734-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8nlm*<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the complex conjugate. This eliminates the occurrences of<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-735-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">i=\u22121<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in the above calculation.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901581153\">Consider an electron in a state of zero angular momentum (<span class=\"MathJax_MathML\" id=\"MathJax-Element-736-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>). In this case, the electron\u2019s wave function depends only on the radial coordinate<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em>. (Refer to the states<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-737-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-738-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200<\/span><\/span><\/span><span>\u00a0<\/span>in<span>\u00a0<\/span>Table 4.1.) The infinitesimal volume element corresponds to a spherical shell of radius<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>and infinitesimal thickness<span>\u00a0<\/span><em data-effect=\"italics\">dr<\/em>, written as<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902923744\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-739-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-739-Frame\"><span class=\"MathJax_MathContainer\"><span>dV=4\u03c0r2dr.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.10]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The probability of finding the electron in the region<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-740-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r+dr<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(\u201cat approximately<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u201d) is<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-741-Frame\"><span class=\"MathJax_MathContainer\"><span>P(r)dr=|\u03c8n00|24\u03c0r2dr.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.11]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Here<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">P<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">) is called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term336\" style=\"font-size: 14pt\">radial probability density function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(a probability per unit length). For an electron in the ground state of hydrogen, the probability of finding an electron in the region<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-742-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r+dr<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902719337\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-743-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-743-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8n00|24\u03c0r2dr=(4\/a03)r2exp(\u22122r\/a0)dr,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.12]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-744-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">a0=0.5<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">angstroms. The radial probability density function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">P<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">) is plotted in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.7. The area under the curve between any two radial positions, say<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-745-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r1<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-746-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r2<\/span><\/span><span style=\"font-size: 14pt\">, gives the probability of finding the electron in that radial range. To find the most probable radial position, we set the first derivative of this function to zero (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-747-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">dP\/dr=0<\/span><\/span><span style=\"font-size: 14pt\">) and solve for<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">. The most probable radial position is not equal to the average or expectation value of the radial position because<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-748-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8n00|2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is not symmetrical about its peak value.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_RadProb\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"452\"]<img alt=\"A graph of the function P of r as a function of r is shown. It is zero at r = 0, rises to a maximum at r = a sub 0, then gradually decreases and goes asymptotically to zero at large r. The maximum is at the most probable radial position. The area of the region under the curve from r sub 1 to r sub 2 is shaded.\" data-media-type=\"image\/jpeg\" id=\"99142\" src=\"https:\/\/cnx.org\/resources\/d575d3f19079d9f4eca80ca0a2a30961bdb92d4f\" width=\"452\" height=\"309\" \/> Figure 4.7 The radial probability density function for the ground state of hydrogen.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170902650156\">If the electron has orbital angular momentum (<span class=\"MathJax_MathML\" id=\"MathJax-Element-749-Frame\"><span class=\"MathJax_MathContainer\"><span>l\u22600<\/span><\/span><\/span>), then the wave functions representing the electron depend on the angles<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-750-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-751-Frame\"><span class=\"MathJax_MathContainer\"><span><\/span><\/span><\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-752-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5;<\/span><\/span><\/span>that is,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-753-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm=<\/span><\/span><\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-754-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm<\/span><\/span><\/span><span>\u00a0<\/span>(<em data-effect=\"italics\">r<\/em>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-755-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-756-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span>). Atomic orbitals for three states with<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-757-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-758-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>are shown in<span>\u00a0<\/span>Figure 4.8. An<span>\u00a0<\/span><span data-type=\"term\" id=\"term337\">atomic orbital<\/span><span>\u00a0<\/span>is a region in space that encloses a certain percentage (usually 90%) of the electron probability. (Sometimes atomic orbitals are referred to as \u201cclouds\u201d of probability.) Notice that these distributions are pronounced in certain directions. This directionality is important to chemists when they analyze how atoms are bound together to form molecules.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_EOrbitals\"><span data-alt=\"This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.\" data-type=\"media\" id=\"fs-id1170902879204\"><img alt=\"This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.\" data-media-type=\"image\/jpeg\" id=\"50274\" src=\"https:\/\/cnx.org\/resources\/a926205e2c9fd931649b35348275925c59d49639\" \/><\/span><\/figure>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Figure\u00a04<\/span><span class=\"os-number\">.8<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-divider\"><\/span><span class=\"os-caption\">The probability density distributions for three states with\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-759-Frame\"><span class=\"MathJax_MathContainer\">n=2<\/span><\/span>\u00a0and\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-760-Frame\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span>. The distributions are directed along the (a)\u00a0x-axis, (b)\u00a0y-axis, and (c)\u00a0z-axis.<\/span><\/em><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902890222\">A slightly different representation of the wave function is given in<span>\u00a0<\/span>Figure 4.9. In this case, light and dark regions indicate locations of relatively high and low probability, respectively. In contrast to the Bohr model of the hydrogen atom, the electron does not move around the proton nucleus in a well-defined path. Indeed, the uncertainty principle makes it impossible to know how the electron gets from one place to another.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_01_PClouds\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"633\"]<img alt=\"The figure shows probability clouds for electrons in the n equals 1, 2 and 3, l equals 0, 1 and 2 states in a 3 by 3 grid. n=1, l=0 is a spherically symmetric distribution, brighter in the center and gradually fading with increasing radius, with no nodes. n=2, l=0 is a spherically symmetric distribution with a spherical, concentric node. The node appears as a black circle within the cloud. The cloud is brightest in the center, fading to black at the node, brightening again to outside the node (but not as bright as at the center of the cloud), then fading again at large r. n=2, l=1 has a planar node along the diameter of the cloud, appearing as a dark line across the distribution and indentations at the edges. The cloud is brightest near the center, above and below the node. n=3, l=0 is a spherically symmetric distribution with two spherical, concentric nodes. The nodes appear as concentric black circles within the cloud. The cloud is brightest in the center, fading to black at the first node, brightening again to a maximum brightness outside the node, fading to black at the second node brightening again, then fading again at large r. The local maxima (at the center, between the nodes, and outside the outer node) decrease in intensity. n=3, l=2 has both a concentric circular node and a planar node along the diameter, appearing as a circle in and line across the cloud. The cloud is brightest inside the circular node. A second local maximum brightness is seen within the lobes above and below the planar node. n=3, l=2 has two planar nodes, which appear as an X across the cloud. The quarters of the cloud thus defined are deeply indented at the edges, forming rounded lobes. The cloud is brightest near the center.\" data-media-type=\"image\/jpeg\" id=\"69215\" src=\"https:\/\/cnx.org\/resources\/ab635220c16ef185ce51a8eccf05d20c3722cac2\" width=\"633\" height=\"638\" \/> Figure 4.9 Probability clouds for the electron in the ground state and several excited states of hydrogen. The probability of finding the electron is indicated by the shade of color; the lighter the coloring, the greater the chance of finding the electron.[\/caption]<\/figure>\r\n<div>\r\n\r\n&nbsp;\r\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"<div data-type=\"abstract\" id=\"45781\" class=\"ui-has-child-title\">\n<header>\n<div class=\"textbox textbox--learning-objectives\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe the hydrogen atom in terms of wave function, probability density, total energy, and orbital angular momentum<\/li>\n<li>Identify the physical significance of each of the quantum numbers (<span class=\"MathJax_MathML\" id=\"MathJax-Element-577-Frame\"><span class=\"MathJax_MathContainer\"><span>n,l,m<\/span><\/span><\/span>) of the hydrogen atom<\/li>\n<li>Distinguish between the Bohr and Schr\u00f6dinger models of the atom<\/li>\n<li>Use quantum numbers to calculate important information about the hydrogen atom<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The hydrogen atom is the simplest atom in nature and, therefore, a good starting point to study atoms and atomic structure. The hydrogen atom consists of a single negatively charged electron that moves about a positively charged proton (<\/span>Figure 4.2<span style=\"font-size: 14pt\">). In Bohr\u2019s model, the electron is pulled around the proton in a perfectly circular orbit by an attractive Coulomb force. The proton is approximately 1800 times more massive than the electron, so the proton moves very little in response to the force on the proton by the electron. (This is analogous to the Earth-Sun system, where the Sun moves very little in response to the force exerted on it by Earth.) An explanation of this effect using Newton\u2019s laws is given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:fdd4b413-6910-44d5-801d-0f4223bc7a31@5\" data-page=\"48\" style=\"font-size: 14pt\">Photons and Matter Waves<\/a><span style=\"font-size: 14pt\">.<\/span><\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_HydAtom\">\n<figure style=\"width: 429px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The Bohr model of the hydrogen atom has the proton, charge q = plus e, at the center and the electron, charge q = minus e, in a circular orbit centered on the proton.\" data-media-type=\"image\/jpeg\" id=\"57666\" src=\"https:\/\/cnx.org\/resources\/a5587a5541704e45cfdf7c6152d84a08d4a569e2\" width=\"429\" height=\"275\" \/><figcaption class=\"wp-caption-text\">Figure 4.2 A representation of the Bohr model of the hydrogen atom.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\">With the assumption of a fixed proton, we focus on the motion of the electron.<\/div>\n<\/div>\n<p>In the electric field of the proton, the potential energy of the electron is<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903082088\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-578-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-578-Frame\"><span class=\"MathJax_MathContainer\"><span>U(r)=\u2212ke2r,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.1]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-579-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">k=1\/4\u03c0\u03b50<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the distance between the electron and the proton. As we saw earlier, the force on an object is equal to the negative of the gradient (or slope) of the potential energy function. For the special case of a hydrogen atom, the force between the electron and proton is an attractive Coulomb force.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902719456\">Notice that the potential energy function<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>) does not vary in time. As a result,<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term329\">Schr\u00f6dinger\u2019s equation<\/span><span>\u00a0<\/span>of the hydrogen atom reduces to two simpler equations: one that depends only on space (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>) and another that depends only on time (<em data-effect=\"italics\">t<\/em>). (The separation of a wave function into space- and time-dependent parts for time-independent potential energy functions is discussed in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:018ccaf5-d217-467a-b03a-d2ea9d46f6ee\" data-page=\"56\">Quantum Mechanics<\/a>.) We are most interested in the space-dependent equation:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903052286\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-580-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-580-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22me(\u22022\u03c8\u2202x2+\u22022\u03c8\u2202y2+\u22022\u03c8\u2202z2)\u2212ke2r\u03c8=E\u03c8,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.2]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-581-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8=\u03c8(x,y,z)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the three-dimensional wave function of the electron,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-582-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">me<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the mass of the electron, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the total energy of the electron. Recall that the total wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-583-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,y,z,t),<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the product of the space-dependent wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-584-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8=\u03c8(x,y,z)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the time-dependent wave function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-585-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c6=\u03c6(t)<\/span><\/span><span style=\"font-size: 14pt\">.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902893842\">In addition to being time-independent,<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>) is also spherically symmetrical. This suggests that we may solve Schr\u00f6dinger\u2019s equation more easily if we express it in terms of the spherical coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-586-Frame\"><span class=\"MathJax_MathContainer\"><span>(r,\u03b8,\u03d5)<\/span><\/span><\/span><span>\u00a0<\/span>instead of rectangular coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-587-Frame\"><span class=\"MathJax_MathContainer\"><span>(x,y,z)<\/span><\/span><\/span>. A spherical coordinate system is shown in<span>\u00a0<\/span>Figure 4.3. In spherical coordinates, the variable<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>is the radial coordinate,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-588-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>is the polar angle (relative to the vertical<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis), and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-589-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>is the azimuthal angle (relative to the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>-axis). The relationship between spherical and rectangular coordinates is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-590-Frame\"><span class=\"MathJax_MathContainer\"><span>x=rsin\u03b8cos\u03d5,y=rsin\u03b8sin\u03d5,z=rcos\u03b8.<\/span><\/span><\/span><\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_SphCoord\">\n<figure style=\"width: 418px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"An x y z coordinate system is shown, along with a point P and the vector r from the origin to P. In this figure, the point P has positive x, y, and z coordinates. The vector r is inclined by an angle theta from the positive z axis. Its projection on the x y plane makes an angle theta from the positive x axis toward the positive y axis.\" data-media-type=\"image\/jpeg\" id=\"80779\" src=\"https:\/\/cnx.org\/resources\/233c5feaf72800977965229aeb9225feadaab04b\" width=\"418\" height=\"379\" \/><figcaption class=\"wp-caption-text\">Figure 4.3 The relationship between the spherical and rectangular coordinate systems.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170903095566\">The factor<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-591-Frame\"><span class=\"MathJax_MathContainer\"><span>rsin\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>is the magnitude of a vector formed by the projection of the polar vector onto the<span>\u00a0<\/span><em data-effect=\"italics\">xy<\/em>-plane. Also, the coordinates of<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em><span>\u00a0<\/span>are obtained by projecting this vector onto the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>&#8211; and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>-axes, respectively. The inverse transformation gives<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903139998\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-592-Frame\"><span class=\"MathJax_MathContainer\"><span>r=x2+y2+z2,\u03b8=cos\u22121(zr),\u03d5=cos\u22121(xx2+y2).<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170903044398\">Schr\u00f6dinger\u2019s wave equation for the hydrogen atom in spherical coordinates is discussed in more advanced courses in modern physics, so we do not consider it in detail here. However, due to the spherical symmetry of<span>\u00a0<\/span><em data-effect=\"italics\">U<\/em>(<em data-effect=\"italics\">r<\/em>), this equation reduces to three simpler equations: one for each of the three coordinates<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-593-Frame\"><span class=\"MathJax_MathContainer\"><span>(r,\u03b8,and\u03d5).<\/span><\/span><\/span><span>\u00a0<\/span>Solutions to the time-independent wave function are written as a product of three functions:<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902913459\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-594-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(r,\u03b8,\u03d5)=R(r)\u0398(\u03b8)\u03a6(\u03d5),<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902898644\">where<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>is the radial function dependent on the radial coordinate<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>only;<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-595-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0398<\/span><\/span><\/span><span>\u00a0<\/span>is the polar function dependent on the polar coordinate<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-596-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>only; and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-597-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a6<\/span><\/span><\/span><span>\u00a0<\/span>is the phi function of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-598-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>only. Valid solutions to Schr\u00f6dinger\u2019s equation<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-599-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(r,\u03b8,\u03d5)<\/span><\/span><\/span><span>\u00a0<\/span>are labeled by the quantum numbers<span>\u00a0<\/span><em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>.<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901533514\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-600-Frame\"><span class=\"MathJax_MathContainer\"><span>n:principal quantum number l:angular momentum quantum number m:angular momentum projection quantum number<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902666863\">(The reasons for these names will be explained in the next section.) The radial function<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">n<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>; the polar function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-601-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0398<\/span><\/span><\/span><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>; and the phi function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-602-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a6<\/span><\/span><\/span><span>\u00a0<\/span>depends only on<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>. The dependence of each function on quantum numbers is indicated with subscripts:<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901741972\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-603-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm(r,\u03b8,\u03d5)=Rnl(r)\u0398lm(\u03b8)\u03a6m(\u03d5).<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902925786\">Not all sets of quantum numbers (<em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>) are possible. For example, the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can never be greater or equal to the principal quantum number<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-604-Frame\"><span class=\"MathJax_MathContainer\"><span>n(l&lt;n)<\/span><\/span><\/span>. Specifically, we have<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902660979\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-605-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,2,3,\u2026l=0,1,2,\u2026,(n\u22121)m=\u2212l,(\u2212l+1),\u2026,0,\u2026,(+l\u22121),+l<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901578315\">Notice that for the ground state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-606-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-607-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-608-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0<\/span><\/span><\/span>. In other words, there is only one quantum state with the wave function for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-609-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span>, and it is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-610-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100<\/span><\/span><\/span>. However, for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-611-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>, we have<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902941717\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-612-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0,m=0l=1,m=\u22121,0,1.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901800555\">Therefore, the allowed states for the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-613-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><span>\u00a0<\/span>state are<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-614-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-615-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c821\u22121,\u03c8210<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-616-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8211<\/span><\/span><\/span>. Example wave functions for the hydrogen atom are given in<span>\u00a0<\/span>Table 4.1. Note that some of these expressions contain the letter<span>\u00a0<\/span><em data-effect=\"italics\">i<\/em>, which represents<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-617-Frame\"><span class=\"MathJax_MathContainer\"><span>\u22121<\/span><\/span><\/span>. When probabilities are calculated, these complex numbers do not appear in the final answer.<\/p>\n<div class=\"os-table\">\n<table id=\"fs-id1170901599432\" summary=\"Table 8.1 Wave Functions of the Hydrogen Atom\">\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-618-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,l=0,ml=0<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-619-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100=1\u03c01a03\/2e\u2212r\/a0<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-620-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=0,ml=0<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-621-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200=142\u03c01a03\/2(2\u2212ra0)e\u2212r\/2a0<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-622-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=\u22121<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-623-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c821\u22121=18\u03c01a03\/2ra0e\u2212r\/2a0sin\u03b8e\u2212i\u03d5<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-624-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=0<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-625-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8210=142\u03c01a03\/2ra0e\u2212r\/2a0cos\u03b8<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-626-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2,l=1,ml=1<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-627-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8211=18\u03c01a03\/2ra0e\u2212r\/2a0sin\u03b8ei\u03d5<\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Wave Functions of the Hydrogen Atom<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\n<\/div>\n<section id=\"fs-id1170902884504\" data-depth=\"1\">\n<h3 data-type=\"title\">Physical Significance of the Quantum Numbers<\/h3>\n<p id=\"fs-id1170902884509\">Each of the three quantum numbers of the hydrogen atom (<em data-effect=\"italics\">n<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>) is associated with a different physical quantity. The<span>\u00a0<\/span><span data-type=\"term\" id=\"term330\">principal quantum number<\/span><span>\u00a0<\/span><em data-effect=\"italics\">n<\/em><span>\u00a0<\/span>is associated with the total energy of the electron,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-628-Frame\"><span class=\"MathJax_MathContainer\"><span>En<\/span><\/span><\/span>. According to Schr\u00f6dinger\u2019s equation:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902869741\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-629-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-629-Frame\"><span class=\"MathJax_MathContainer\"><span>En=\u2212(mek2e422)(1n2)=\u2212E0(1n2),<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.3]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-630-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E0=\u221213.6eV.<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Notice that this expression is identical to that of Bohr\u2019s model. As in the Bohr model, the electron in a particular state of energy does not radiate.<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1170902858591\" class=\"ui-has-child-title\">\n<header><\/header>\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\"><\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170902858593\"><span data-type=\"title\"><strong>How Many Possible States?<\/strong><\/span><\/p>\n<p>For the hydrogen atom, how many possible quantum states correspond to the principal number<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-631-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>? What are the energies of these states?<\/p>\n<p><span data-type=\"title\"><strong>Strategy<\/strong><\/span><\/p>\n<p>For a hydrogen atom of a given energy, the number of allowed states depends on its orbital angular momentum. We can count these states for each value of the principal quantum number,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-632-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1,2,3.<\/span><\/span><\/span><span>\u00a0<\/span>However, the total energy depends on the principal quantum number only, which means that we can use<span>\u00a0<\/span>Equation 4.3<span>\u00a0<\/span>and the number of states counted.<\/p>\n<p><span data-type=\"title\"><strong>Solution<\/strong><\/span><\/p>\n<p>If<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-633-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>, the allowed values of<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>are 0, 1, and 2. If<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-634-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-635-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0<\/span><\/span><\/span><span>\u00a0<\/span>(1 state). If<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-636-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-637-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u22121,0,+1<\/span><\/span><\/span><span>\u00a0<\/span>(3 states); and if<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-638-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-639-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u22122,\u22121,0,+1,+2<\/span><\/span><\/span><span>\u00a0<\/span>(5 states). In total, there are<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-640-Frame\"><span class=\"MathJax_MathContainer\"><span>1+3+5=9<\/span><\/span><\/span><span>\u00a0<\/span>allowed states. Because the total energy depends only on the principal quantum number,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-641-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>, the energy of each of these states is<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901632559\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-642-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>En3=\u2212E0(1n2)=\u221213.6eV9=\u22121.51eV.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">An electron in a hydrogen atom can occupy many different angular momentum states with the very same energy. As the orbital angular momentum increases, the number of the allowed states with the same energy increases.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p><span style=\"font-size: 14pt\">The<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term331\" style=\"font-size: 14pt\">angular momentum orbital quantum number<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">l<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is associated with the orbital angular momentum of the electron in a hydrogen atom. Quantum theory tells us that when the hydrogen atom is in the state<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-643-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8nlm<\/span><\/span><span style=\"font-size: 14pt\">, the magnitude of its orbital angular momentum is<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-644-Frame\"><span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210f,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.4]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><\/p>\n<\/section>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901842631\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-645-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0,1,2,\u2026,(n\u22121).<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170903077401\">This result is slightly different from that found with Bohr\u2019s theory, which quantizes angular momentum according to the rule<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-646-Frame\"><span class=\"MathJax_MathContainer\"><span>L=n,where n=1,2,3,&#8230;.<\/span><\/span><\/span><\/p>\n<p id=\"fs-id1170901596724\">Quantum states with different values of orbital angular momentum are distinguished using<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term332\">spectroscopic notation<\/span><span>\u00a0<\/span>(Table 4.2). The designations<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">f<\/em><span>\u00a0<\/span>result from early historical attempts to classify atomic spectral lines. (The letters stand for sharp, principal, diffuse, and fundamental, respectively.) After<span>\u00a0<\/span><em data-effect=\"italics\">f<\/em>, the letters continue alphabetically.<\/p>\n<p id=\"fs-id1170902899390\">The<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term333\">ground state<\/span><span>\u00a0<\/span>of hydrogen is designated as the 1<em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state, where \u201c1\u201d indicates the energy level<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-647-Frame\"><span class=\"MathJax_MathContainer\"><span>(n=1)<\/span><\/span><\/span><span>\u00a0<\/span>and \u201c<em data-effect=\"italics\">s<\/em>\u201d indicates the orbital angular momentum state (<span class=\"MathJax_MathML\" id=\"MathJax-Element-648-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>). When<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-649-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can be either 0 or 1. The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-650-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-651-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><span>\u00a0<\/span>state is designated \u201c2<em data-effect=\"italics\">s<\/em>.\u201d The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-652-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-653-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>state is designated \u201c2<em data-effect=\"italics\">p<\/em>.\u201d When<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-654-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><span>\u00a0<\/span>can be 0, 1, or 2, and the states are 3<em data-effect=\"italics\">s<\/em>, 3<em data-effect=\"italics\">p<\/em>, and 3<em data-effect=\"italics\">d<\/em>, respectively. Notation for other quantum states is given in<span>\u00a0<\/span>Table 4.3.<\/p>\n<p id=\"fs-id1170902891111\">The<span>\u00a0<\/span><span data-type=\"term\" id=\"term334\">angular momentum projection quantum number<\/span><span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>is associated with the azimuthal angle<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-655-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>(see<span>\u00a0<\/span>Figure 4.3) and is related to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of orbital angular momentum of an electron in a hydrogen atom. This component is given by<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-656-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz=m\u210f,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.5]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901867971\">where<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901621612\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-657-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u2212l,\u2212l+1,\u2026,0,\u2026,+l\u22121,l.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902849835\">The<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of angular momentum is related to the magnitude of angular momentum by<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903125406\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-658-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-658-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz=Lcos\u03b8,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.6]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-659-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the angle between the angular momentum vector and the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis. Note that the direction of the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis is determined by experiment\u2014that is, along any direction, the experimenter decides to measure the angular momentum. For example, the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-direction might correspond to the direction of an external magnetic field. The relationship between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-660-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">Lz and L<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.4.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_ZcompAng\">\n<figure style=\"width: 387px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"An x y z coordinate system is shown. The vector L is at an angle theta to the positive z axis and has positive z component L sub z equal to m times h bar. The x and y components are positive but not specified.\" data-media-type=\"image\/jpeg\" id=\"63328\" src=\"https:\/\/cnx.org\/resources\/688e44bcedab2fdb4d80de7eeb83c8e20c1b9232\" width=\"387\" height=\"355\" \/><figcaption class=\"wp-caption-text\">Figure 4.4 The z-component of angular momentum is quantized with its own quantum number m.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<div class=\"os-table\">\n<table id=\"fs-id1170902779373\" summary=\"Table 8.2 Spectroscopic Notation and Orbital Angular Momentum\">\n<thead>\n<tr valign=\"top\">\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Orbital Quantum Number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em><\/th>\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Angular Momentum<\/th>\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">State<\/th>\n<th scope=\"col\" data-align=\"left\" data-valign=\"top\">Spectroscopic Name<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">0<\/td>\n<td data-align=\"left\" data-valign=\"top\">0<\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">Sharp<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">1<\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-661-Frame\"><span class=\"MathJax_MathContainer\"><span>2h<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">Principal<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">2<\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-662-Frame\"><span class=\"MathJax_MathContainer\"><span>6h<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">d<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">Diffuse<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">3<\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-663-Frame\"><span class=\"MathJax_MathContainer\"><span>12h<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">f<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">Fundamental<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">4<\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-664-Frame\"><span class=\"MathJax_MathContainer\"><span>20h<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">g<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\">5<\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-665-Frame\"><span class=\"MathJax_MathContainer\"><span>30h<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><em data-effect=\"italics\">h<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Spectroscopic Notation and Orbital Angular Momentum<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\n<\/div>\n<div class=\"os-table\">\n<table id=\"fs-id1170903078180\" summary=\"Table 8.3 Spectroscopic Description of Quantum States\">\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-666-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-667-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-668-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-669-Frame\"><span class=\"MathJax_MathContainer\"><span>l=3<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-670-Frame\"><span class=\"MathJax_MathContainer\"><span>l=4<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-671-Frame\"><span class=\"MathJax_MathContainer\"><span>l=5<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-672-Frame\"><span class=\"MathJax_MathContainer\"><span>n=1<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">1<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-673-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">2<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">2<em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-674-Frame\"><span class=\"MathJax_MathContainer\"><span>n=3<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">3<em data-effect=\"italics\">d<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-675-Frame\"><span class=\"MathJax_MathContainer\"><span>n=4<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">d<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">4<em data-effect=\"italics\">f<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-676-Frame\"><span class=\"MathJax_MathContainer\"><span>n=5<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">d<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">f<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">5<em data-effect=\"italics\">g<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><span class=\"MathJax_MathML\" id=\"MathJax-Element-677-Frame\"><span class=\"MathJax_MathContainer\"><span>n=6<\/span><\/span><\/span><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">s<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">p<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">d<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">f<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">g<\/em><\/td>\n<td data-align=\"left\" data-valign=\"top\">6<em data-effect=\"italics\">h<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Table 4<\/span><span class=\"os-number\">.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-title\" data-type=\"title\">Spectroscopic Description of Quantum States<\/span><\/em><span class=\"os-divider\"><\/span><span class=\"os-caption\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901531308\">The quantization of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-678-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz<\/span><\/span><\/span><span>\u00a0<\/span>is equivalent to the quantization of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-679-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>. Substituting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-680-Frame\"><span class=\"MathJax_MathContainer\"><span>l(l+1)\u210f<\/span><\/span><\/span><span>\u00a0<\/span>for<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-681-Frame\"><span class=\"MathJax_MathContainer\"><span>Lz<\/span><\/span><\/span><span>\u00a0<\/span>into this equation, we find<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902680809\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-682-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-682-Frame\"><span class=\"MathJax_MathContainer\"><span>m\u210f=l(l+1)\u210fcos\u03b8.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.7]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Thus, the angle<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-683-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is quantized with the particular values<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901754937\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-684-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-684-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8=cos\u22121(ml(l+1)).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.8]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Notice that both the polar angle (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-685-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b8<\/span><\/span><span style=\"font-size: 14pt\">) and the projection of the angular momentum vector onto an arbitrary<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-axis (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-686-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"font-size: 14pt\">) are quantized.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170901648045\">The quantization of the polar angle for the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-687-Frame\"><span class=\"MathJax_MathContainer\"><span>l=3<\/span><\/span><\/span><span>\u00a0<\/span>state is shown in<span>\u00a0<\/span>Figure 4.5. The orbital angular momentum vector lies somewhere on the surface of a cone with an opening angle<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-688-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>relative to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis (unless<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-689-Frame\"><span class=\"MathJax_MathContainer\"><span>m=0,<\/span><\/span><\/span><span>\u00a0<\/span>in which case<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-690-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8=90\u00b0<\/span><\/span><\/span><span>\u00a0<\/span>and the vector points are perpendicular to the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis).<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_QuantTheta\">\n<figure style=\"width: 349px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Seven vector, all of the same length L, are drawn at 7 different angles to the z axis. The z components of the vectors are indicated both by horizontal lines from the tip of the vector to the z axis and by labels on the z axis. For four of the vectors, the angle between the z axis and the vector is also labeled. The z component values are 3 h bar at angle theta sub three, 2 h bar at angle theta sub two, h bar at angle theta sub one, zero at angle theta sub zero, minus h bar, minus 2 h bar, and minus 3 h bar.\" data-media-type=\"image\/jpeg\" id=\"77046\" src=\"https:\/\/cnx.org\/resources\/0702c75c508115325b2f61409dae642d3dfca9df\" width=\"349\" height=\"496\" \/><figcaption class=\"wp-caption-text\">Figure 4.5 The quantization of orbital angular momentum. Each vector lies on the surface of a cone with axis along the z-axis.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170901577555\">A detailed study of angular momentum reveals that we cannot know all three components simultaneously. In the previous section, the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-component of orbital angular momentum has definite values that depend on the quantum number<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em>. This implies that we cannot know both<span>\u00a0<\/span><em data-effect=\"italics\">x-<\/em><span>\u00a0<\/span>and<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>-components of angular momentum,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-691-Frame\"><span class=\"MathJax_MathContainer\"><span>Lx<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-692-Frame\"><span class=\"MathJax_MathContainer\"><span>Ly<\/span><\/span><\/span>, with certainty. As a result, the precise direction of the orbital angular momentum vector is unknown.<\/p>\n<div data-type=\"example\" id=\"fs-id1170902705884\" class=\"ui-has-child-title\">\n<header><\/header>\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.2<\/span><span class=\"os-divider\"><\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170902705886\"><span data-type=\"title\"><strong>What Are the Allowed Directions?<\/strong><\/span><\/p>\n<p>Calculate the angles that the angular momentum vector<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-693-Frame\"><span class=\"MathJax_MathContainer\"><span>L\u2192<\/span><\/span><\/span><span>\u00a0<\/span>can make with the<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-axis for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-694-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span>, as shown in<span>\u00a0<\/span>Figure 4.6.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_AngMoment\">\n<figure style=\"width: 431px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The image shows three possible values of a component of a given angular momentum along z-axis. The upper circular orbit is shown for m sub t = 1 at a distance L sub z above the origin. The vector L makes an angle of theta one with the z axis. The radius of the orbit is the component of L perpendicular to the z axis. The middle circular orbit is shown for m sub t = 0. It is in the x y plane. The vector L makes an angle of theta two of 90 degrees with the z axis. The radius of the orbit is L. The lower circular orbit is shown for m sub t = -1 at a distance L sub z below the origin. The vector L makes an angle of theta three with the z axis. The radius of the orbit is the component of L perpendicular to the z axis.\" data-media-type=\"image\/jpeg\" id=\"64105\" src=\"https:\/\/cnx.org\/resources\/8eab941cffacd2c77a675ec48b93cf415e951c33\" width=\"431\" height=\"571\" \/><figcaption class=\"wp-caption-text\">Figure 4.6 The component of a given angular momentum along the z-axis (defined by the direction of a magnetic field) can have only certain values. These are shown here for l=1, for which m=\u22121,0,and+1. The direction of L\u2192 is quantized in the sense that it can have only certain angles relative to the z-axis.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><br \/>\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span><\/span><\/span><\/span>&nbsp;<\/p>\n<p><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">The vectors<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-698-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-699-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192z<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">(in the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction) form a right triangle, where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-700-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the hypotenuse and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-701-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192z<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the adjacent side. The ratio of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-702-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to |<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-703-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">| is the cosine of the angle of interest. The magnitudes<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-704-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L=|L\u2192|<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-705-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">are given by<\/span><br \/>\n<\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902866146\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-706-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210fandLz=m\u210f.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">We are given<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-707-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">ml<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can be<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-708-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">+1,0,or\u22121.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">Thus,<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">has the value given by<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902861390\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-709-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>L=l(l+1)\u210f=2\u210f.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The quantity<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-710-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can have three values, given by<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-711-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz=ml\u210f<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902897710\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-712-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>Lz=ml\u210f={\u210f,ml=+10,ml=0\u2212\u210f,ml=\u22121<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">As you can see in<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span>Figure 4.6<span style=\"text-indent: 1em;font-size: 1rem\">,<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-713-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b8=Lz\/L,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">so for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-714-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">m=+1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, we have<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902923917\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-715-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>cos\u03b81=LZL=\u210f2\u210f=12=0.707.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Thus,<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902652024\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-716-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03b81=cos\u221210.707=45.0\u00b0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Similarly, for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-717-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">m=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, we find<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-718-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b82=0;<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">this gives<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901489064\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-719-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03b82=cos\u221210=90.0\u00b0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Then for<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-720-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml=\u22121<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902765413\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-721-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>cos\u03b83=LZL=\u2212\u210f2\u210f=\u221212=\u22120.707,<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">so that<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902922844\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-722-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03b83=cos\u22121(\u22120.707)=135.0\u00b0.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The angles are consistent with the figure. Only the angle relative to the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis is quantized.<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">can point in any direction as long as it makes the proper angle with the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis. Thus, the angular momentum vectors lie on cones, as illustrated. To see how the correspondence principle holds here, consider that the smallest angle (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-723-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03b81<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">in the example) is for the maximum value of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-724-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">namely<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-725-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml=l.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">For that smallest angle,<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901642419\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-726-Frame\"><span class=\"MathJax_MathContainer\"><span>cos\u03b8=LzL=ll(l+1),<\/p>\n<p><\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">which approaches 1 as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">becomes very large. If<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-727-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">cos\u03b8=1<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, then<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-728-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03b8=0\u00ba<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. Furthermore, for large<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">, there are many values of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-729-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">ml<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so that all angles become possible as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">l<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">gets very large.<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1170901751543\" data-depth=\"1\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a04<\/span><\/span><span class=\"os-number\">.1<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header><span style=\"font-size: 1rem\">Can the magnitude of<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-730-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">Lz<\/span><\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">ever be equal to<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">?<\/span><\/header>\n<\/div>\n<\/div>\n<h3 data-type=\"title\">Using the Wave Function to Make Predictions<\/h3>\n<p id=\"fs-id1170901751548\">As we saw earlier, we can use quantum mechanics to make predictions about physical events by the use of probability statements. It is therefore proper to state, \u201cAn electron is located within this volume with this probability at this time,\u201d but not, \u201cAn electron is located at the position (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">y<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>) at this time.\u201d To determine the probability of finding an electron in a hydrogen atom in a particular region of space, it is necessary to integrate the probability density<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-731-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8nlm|2<\/span><\/span><\/span><span>\u00a0<\/span>over that region:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902682281\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-732-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-732-Frame\"><span class=\"MathJax_MathContainer\"><span>Probability=\u222bvolume|\u03c8nlm|2dV,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.9]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dV<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is an infinitesimal volume element. If this integral is computed for all space, the result is 1, because the probability of the particle to be located<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">somewhere<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is 100% (the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term335\" style=\"font-size: 14pt\">normalization<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">condition). In a more advanced course on modern physics, you will find that<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-733-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8nlm|2=\u03c8nlm*\u03c8nlm,<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-734-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8nlm*<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the complex conjugate. This eliminates the occurrences of<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-735-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">i=\u22121<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in the above calculation.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170901581153\">Consider an electron in a state of zero angular momentum (<span class=\"MathJax_MathML\" id=\"MathJax-Element-736-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span>). In this case, the electron\u2019s wave function depends only on the radial coordinate<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em>. (Refer to the states<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-737-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8100<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-738-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8200<\/span><\/span><\/span><span>\u00a0<\/span>in<span>\u00a0<\/span>Table 4.1.) The infinitesimal volume element corresponds to a spherical shell of radius<span>\u00a0<\/span><em data-effect=\"italics\">r<\/em><span>\u00a0<\/span>and infinitesimal thickness<span>\u00a0<\/span><em data-effect=\"italics\">dr<\/em>, written as<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902923744\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-739-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-739-Frame\"><span class=\"MathJax_MathContainer\"><span>dV=4\u03c0r2dr.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.10]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The probability of finding the electron in the region<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-740-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r+dr<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(\u201cat approximately<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u201d) is<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-741-Frame\"><span class=\"MathJax_MathContainer\"><span>P(r)dr=|\u03c8n00|24\u03c0r2dr.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.11]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Here<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">P<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">) is called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term336\" style=\"font-size: 14pt\">radial probability density function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(a probability per unit length). For an electron in the ground state of hydrogen, the probability of finding an electron in the region<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-742-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r+dr<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902719337\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-743-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-743-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8n00|24\u03c0r2dr=(4\/a03)r2exp(\u22122r\/a0)dr,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.12]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-744-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">a0=0.5<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">angstroms. The radial probability density function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">P<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">) is plotted in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.7. The area under the curve between any two radial positions, say<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-745-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r1<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-746-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r2<\/span><\/span><span style=\"font-size: 14pt\">, gives the probability of finding the electron in that radial range. To find the most probable radial position, we set the first derivative of this function to zero (<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-747-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">dP\/dr=0<\/span><\/span><span style=\"font-size: 14pt\">) and solve for<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">. The most probable radial position is not equal to the average or expectation value of the radial position because<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-748-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8n00|2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is not symmetrical about its peak value.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_RadProb\">\n<figure style=\"width: 452px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"A graph of the function P of r as a function of r is shown. It is zero at r = 0, rises to a maximum at r = a sub 0, then gradually decreases and goes asymptotically to zero at large r. The maximum is at the most probable radial position. The area of the region under the curve from r sub 1 to r sub 2 is shaded.\" data-media-type=\"image\/jpeg\" id=\"99142\" src=\"https:\/\/cnx.org\/resources\/d575d3f19079d9f4eca80ca0a2a30961bdb92d4f\" width=\"452\" height=\"309\" \/><figcaption class=\"wp-caption-text\">Figure 4.7 The radial probability density function for the ground state of hydrogen.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170902650156\">If the electron has orbital angular momentum (<span class=\"MathJax_MathML\" id=\"MathJax-Element-749-Frame\"><span class=\"MathJax_MathContainer\"><span>l\u22600<\/span><\/span><\/span>), then the wave functions representing the electron depend on the angles<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-750-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-751-Frame\"><span class=\"MathJax_MathContainer\"><span><\/span><\/span><\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-752-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5;<\/span><\/span><\/span>that is,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-753-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm=<\/span><\/span><\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-754-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8nlm<\/span><\/span><\/span><span>\u00a0<\/span>(<em data-effect=\"italics\">r<\/em>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-755-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b8<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-756-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span>). Atomic orbitals for three states with<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-757-Frame\"><span class=\"MathJax_MathContainer\"><span>n=2<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-758-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>are shown in<span>\u00a0<\/span>Figure 4.8. An<span>\u00a0<\/span><span data-type=\"term\" id=\"term337\">atomic orbital<\/span><span>\u00a0<\/span>is a region in space that encloses a certain percentage (usually 90%) of the electron probability. (Sometimes atomic orbitals are referred to as \u201cclouds\u201d of probability.) Notice that these distributions are pronounced in certain directions. This directionality is important to chemists when they analyze how atoms are bound together to form molecules.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_EOrbitals\"><span data-alt=\"This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.\" data-type=\"media\" id=\"fs-id1170902879204\"><img decoding=\"async\" alt=\"This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.\" data-media-type=\"image\/jpeg\" id=\"50274\" src=\"https:\/\/cnx.org\/resources\/a926205e2c9fd931649b35348275925c59d49639\" \/><\/span><\/figure>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\">Figure\u00a04<\/span><span class=\"os-number\">.8<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-divider\"><\/span><span class=\"os-caption\">The probability density distributions for three states with\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-759-Frame\"><span class=\"MathJax_MathContainer\">n=2<\/span><\/span>\u00a0and\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-760-Frame\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span>. The distributions are directed along the (a)\u00a0x-axis, (b)\u00a0y-axis, and (c)\u00a0z-axis.<\/span><\/em><\/div>\n<\/div>\n<p id=\"fs-id1170902890222\">A slightly different representation of the wave function is given in<span>\u00a0<\/span>Figure 4.9. In this case, light and dark regions indicate locations of relatively high and low probability, respectively. In contrast to the Bohr model of the hydrogen atom, the electron does not move around the proton nucleus in a well-defined path. Indeed, the uncertainty principle makes it impossible to know how the electron gets from one place to another.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_01_PClouds\">\n<figure style=\"width: 633px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The figure shows probability clouds for electrons in the n equals 1, 2 and 3, l equals 0, 1 and 2 states in a 3 by 3 grid. n=1, l=0 is a spherically symmetric distribution, brighter in the center and gradually fading with increasing radius, with no nodes. n=2, l=0 is a spherically symmetric distribution with a spherical, concentric node. The node appears as a black circle within the cloud. The cloud is brightest in the center, fading to black at the node, brightening again to outside the node (but not as bright as at the center of the cloud), then fading again at large r. n=2, l=1 has a planar node along the diameter of the cloud, appearing as a dark line across the distribution and indentations at the edges. The cloud is brightest near the center, above and below the node. n=3, l=0 is a spherically symmetric distribution with two spherical, concentric nodes. The nodes appear as concentric black circles within the cloud. The cloud is brightest in the center, fading to black at the first node, brightening again to a maximum brightness outside the node, fading to black at the second node brightening again, then fading again at large r. The local maxima (at the center, between the nodes, and outside the outer node) decrease in intensity. n=3, l=2 has both a concentric circular node and a planar node along the diameter, appearing as a circle in and line across the cloud. The cloud is brightest inside the circular node. A second local maximum brightness is seen within the lobes above and below the planar node. n=3, l=2 has two planar nodes, which appear as an X across the cloud. The quarters of the cloud thus defined are deeply indented at the edges, forming rounded lobes. The cloud is brightest near the center.\" data-media-type=\"image\/jpeg\" id=\"69215\" src=\"https:\/\/cnx.org\/resources\/ab635220c16ef185ce51a8eccf05d20c3722cac2\" width=\"633\" height=\"638\" \/><figcaption class=\"wp-caption-text\">Figure 4.9 Probability clouds for the electron in the ground state and several excited states of hydrogen. The probability of finding the electron is indicated by the shade of color; the lighter the coloring, the greater the chance of finding the electron.<\/figcaption><\/figure>\n<\/figure>\n<div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":615,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"4. Atomic Structure","pb_subtitle":"4.1 The Hydrogen Atom","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-218","chapter","type-chapter","status-publish","hentry"],"part":213,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/users\/615"}],"version-history":[{"count":8,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/218\/revisions"}],"predecessor-version":[{"id":449,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/218\/revisions\/449"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/parts\/213"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/218\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/media?parent=218"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapter-type?post=218"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/contributor?post=218"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/license?post=218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}