{"id":1654,"date":"2017-10-27T16:32:37","date_gmt":"2017-10-27T16:32:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/quantum-numbers-and-rules\/"},"modified":"2017-11-08T03:27:45","modified_gmt":"2017-11-08T03:27:45","slug":"quantum-numbers-and-rules","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/quantum-numbers-and-rules\/","title":{"raw":"Quantum Numbers and Rules","rendered":"Quantum Numbers and Rules"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define quantum number.<\/li>\n<li>Calculate angle of angular momentum vector with an axis.<\/li>\n<li>Define spin quantum number.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1276239\">Physical characteristics that are quantized\u2014such as energy, charge, and angular momentum\u2014are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of <span data-type=\"term\">quantum numbers<\/span>, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules\u2014all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.<\/p>\n<p id=\"import-auto-id2625656\">The <em data-effect=\"italics\">energy states of bound systems are quantized<\/em>, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as [latex]{E}_{n}\\propto 1\/{n}^{2}[\/latex], where [latex]n=1, 2, 3, ...[\/latex]. We define<br>\n[latex]n[\/latex] to be the principal quantum number that labels the basic states of a system. The lowest-energy state has<br>\n[latex]n=1[\/latex], the first excited state has<br>\n[latex]n=2[\/latex], and so on. Thus the allowed values for the principal quantum number are<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]n=1, 2, 3, ....[\/latex]<\/div>\n<p id=\"import-auto-id1390187\">This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of [latex]n[\/latex], as can other characteristics (such as the orbital radii of the hydrogen atom).<\/p>\n<p id=\"import-auto-id3063639\">The fact that the <em data-effect=\"italics\">magnitude of angular momentum is quantized<\/em> was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum [latex]L[\/latex] can have only the values<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left(l=0, 1, 2, ...,\\phantom{\\rule{0.25em}{0ex}}n-1\\right)\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id3069597\">where [latex]l[\/latex] is defined to be the <span data-type=\"term\" id=\"import-auto-id1432038\">angular momentum quantum number<\/span>. The rule for [latex]l[\/latex] in atoms is given in the parentheses. Given [latex]n[\/latex], the value of [latex]l[\/latex] can be any integer from zero up to [latex]n-1[\/latex]. For example, if [latex]n=4[\/latex], then [latex]l[\/latex] can be 0, 1, 2, or 3.<\/p>\n<p>Note that for [latex]n=1[\/latex], [latex]l[\/latex] can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not<br>\n[latex]h\/2\\pi [\/latex] as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in <a href=\"\/contents\/aedd88f0-3f74-4137-b2f1-67af8b4d7832@3#import-auto-id2396594\" class=\"autogenerated-content\">(Figure)<\/a>. The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg\u2019s uncertainty principle\u2014the electron\u2019s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has [latex]n=2[\/latex], so that [latex]l[\/latex] can be either 0 or 1, according to the rule in [latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }[\/latex] . Similarly, for [latex]n=3[\/latex], [latex]l[\/latex] can be 0, 1, or 2. It is often most convenient to state the value of [latex]l[\/latex], a simple integer, rather than calculating the value of [latex]L[\/latex] from [latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }[\/latex]. For example, for [latex]l=2[\/latex], we see that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]L=\\sqrt{2\\left(2+1\\right)}\\frac{h}{2\\pi }=\\sqrt{6}\\frac{h}{2\\pi }=0\\text{.}\\text{390}h=2\\text{.}\\text{58}\u00d7{\\text{10}}^{-\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot s.[\/latex]<\/div>\n<p>It is much simpler to state [latex]l=2[\/latex].<\/p>\n<p id=\"import-auto-id2992778\">As recognized in the Zeeman effect, the <em data-effect=\"italics\">direction of angular momentum is quantized<\/em>. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the [latex]z[\/latex]-axis, can have only certain values of [latex]{L}_{z}[\/latex]. The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of [latex]{L}_{z}[\/latex]<em data-effect=\"italics\"> are<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{L}_{z}={m}_{l}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left({m}_{l}=-l,-l+1, ...,\\phantom{\\rule{0.25em}{0ex}}-1, 0, 1, ...\\phantom{\\rule{0.25em}{0ex}}l-1,\\phantom{\\rule{0.25em}{0ex}}l\\right)\\text{,}[\/latex]<\/div>\n<p>where [latex]{L}_{z}[\/latex] is the <span data-type=\"term\" id=\"import-auto-id2953765\">[latex]z[\/latex]-component of the angular momentum<\/span> and [latex]{m}_{l}[\/latex] is the angular momentum projection quantum number. The rule in parentheses for the values of [latex]{m}_{l}[\/latex] is that it can range from [latex]-l[\/latex] to [latex]l[\/latex] in steps of one. For example, if [latex]l=2[\/latex], then [latex]{m}_{l}[\/latex] can have the five values \u20132, \u20131, 0, 1, and 2. Each [latex]{m}_{l}[\/latex] corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the [latex]z[\/latex]-component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1996787\">\n<div class=\"bc-figcaption figcaption\">The component of a given angular momentum along the [latex]z[\/latex]-axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for [latex]l=1[\/latex], for which [latex]{m}_{l}=-1, 0, and +1[\/latex]. The direction of [latex]L[\/latex] is quantized in the sense that it can have only certain angles relative to the [latex]z[\/latex]-axis.<\/div>\n<p><span data-type=\"media\" data-alt=\"The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_31_08_00a.jpg\" data-media-type=\"image\/jpg\" alt=\"The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.\" width=\"250\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1419670\">\n<div data-type=\"title\" class=\"title\">What Are the Allowed Directions?<\/div>\n<p id=\"import-auto-id2928454\">Calculate the angles that the angular momentum vector [latex]\\mathbf{\\text{L}}[\/latex]<strong data-effect=\"bold\"> can make with the [latex]z[\/latex]-axis for [latex]l=1[\/latex], as illustrated in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>.<\/strong><\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1350660\"><a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a> represents the vectors  [latex]\\mathbf{\\text{L}}[\/latex] and [latex]{\\mathbf{\\text{L}}}_{z}[\/latex] as usual, with arrows proportional to their magnitudes and pointing in the correct directions. [latex]\\mathbf{\\text{L}}[\/latex]<strong data-effect=\"bold\"> and [latex]{\\mathbf{\\text{L}}}_{z}[\/latex] form a right triangle, with [latex]\\mathbf{L}[\/latex] being the hypotenuse and [latex]{\\mathbf{\\text{L}}}_{z}[\/latex] the adjacent side. This means that the ratio of [latex]{\\mathbf{L}}_{z}[\/latex] to [latex]\\mathbf{L}[\/latex] is the cosine of the angle of interest. We can find [latex]\\mathbf{\\text{L}}[\/latex] and [latex]{\\mathbf{\\text{L}}}_{z}[\/latex] using [latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }[\/latex] and [latex]{L}_{z}=m\\frac{h}{2\\pi }[\/latex].<\/strong><\/p>\n<p id=\"import-auto-id3093903\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1602621\">We are given [latex]l=1[\/latex], so that [latex]{m}_{l}[\/latex] can be +1, 0, or \u22121. Thus [latex]L[\/latex] has the value given by [latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]L=\\frac{\\sqrt{l\\left(l+1\\right)}h}{2\\pi }=\\frac{\\sqrt{2}h}{2\\pi }[\/latex]<\/div>\n<p>[latex]{L}_{z}[\/latex]<em data-effect=\"italics\"> can have three values, given by [latex]{L}_{z}={m}_{l}\\frac{h}{2\\pi }[\/latex].<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-250\">[latex]{L}_{z}={m}_{l}\\frac{h}{2\\pi }=\\left\\{\\begin{array}{cccc}\\phantom{\\rule{0.25em}{0ex}}\\frac{h}{2\\pi },&amp; \\phantom{\\rule{0.25em}{0ex}}{m}_{l}&amp; =&amp; +1\\phantom{\\rule{0.25em}{0ex}}\\\\ \\phantom{\\rule{0.25em}{0ex}}0,\\phantom{\\rule{0.25em}{0ex}}&amp; {m}_{l}&amp; =&amp; 0\\phantom{\\rule{0.25em}{0ex}}\\\\ -\\frac{h}{2\\pi },&amp; \\phantom{\\rule{0.25em}{0ex}}{m}_{l}&amp; =&amp; -1\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1349581\">As can be seen in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>,<br>\n[latex]cos\\phantom{\\rule{0.25em}{0ex}}\\theta ={\\text{L}}_{z}\\text{\/L,}[\/latex]<br>\n and so for [latex]{m}_{l}=+1[\/latex], we have<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-985\">[latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{1}=\\frac{{L}_{Z}}{L}=\\frac{\\frac{h}{2\\pi }}{\\frac{\\sqrt{2}h}{2\\pi }}=\\frac{1}{\\sqrt{2}}=0\\text{.}\\text{707.}[\/latex]<\/div>\n<p id=\"import-auto-id2437142\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{1}={\\text{cos}}^{-1}\\text{0.707}=\\text{45}\\text{.}0\u00ba.[\/latex]<\/div>\n<p id=\"import-auto-id2697885\">Similarly, for [latex]{m}_{l}=0[\/latex], we find [latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{2}=0[\/latex]; thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{2}={\\text{cos}}^{-1}0=\\text{90}\\text{.}0\u00ba.[\/latex]<\/div>\n<p id=\"import-auto-id2655556\">And for [latex]{m}_{l}=-1[\/latex],<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{3}=\\frac{{L}_{Z}}{L}=\\frac{-\\frac{h}{2\\pi }}{\\frac{\\sqrt{2}h}{2\\pi }}=-\\frac{1}{\\sqrt{2}}=-0\\text{.}\\text{707,}[\/latex]<\/div>\n<p id=\"import-auto-id2583456\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{3}={\\text{cos}}^{-1}\\left(-0\\text{.}\\text{707}\\right)=\\text{135}\\text{.}0\u00ba.[\/latex]<\/div>\n<p><strong>Discussion<\/strong><\/p>\n<p>The angles are consistent with the figure. Only the angle relative to the [latex]z[\/latex]-axis is quantized. [latex]L[\/latex] can point in any direction as long as it makes the proper angle with the [latex]z[\/latex]-axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle ([latex]{\\theta }_{\\text{1}}[\/latex] in the example) is for the maximum value of [latex]{m}_{l}=0[\/latex], namely [latex]{m}_{l}=l[\/latex]. For that smallest angle,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta =\\frac{{L}_{z}}{L}=\\frac{l}{\\sqrt{l\\left(l+1\\right)}}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1198202\">which approaches 1 as [latex]l[\/latex] becomes very large. If [latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta =1[\/latex], then <em data-effect=\"italics\">[latex]\\theta =0\u00ba[\/latex]. Furthermore, for large [latex]l[\/latex], there are many values of [latex]{m}_{l}[\/latex], so that all angles become possible as [latex]l[\/latex] gets very large.<\/em><\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1448501\">\n<h1 data-type=\"title\">Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction<\/h1>\n<p id=\"import-auto-id3421186\">There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have <em data-effect=\"italics\">intrinsic spin<\/em>, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the <span data-type=\"term\" id=\"import-auto-id2436972\">magnitude of the intrinsic (internal) spin angular momentum<\/span>, [latex]S[\/latex], of an electron is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]S=\\sqrt{s\\left(s+1\\right)}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\text{(}s=1\/2\\phantom{\\rule{0.25em}{0ex}}\\text{for electrons),}[\/latex]<\/div>\n<p>where [latex]s[\/latex] is defined to be the <span data-type=\"term\" id=\"import-auto-id1871891\">spin quantum number<\/span>. This is very similar to the quantization of [latex]L[\/latex] given in [latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }[\/latex], except that the only value allowed for [latex]s[\/latex] for electrons is 1\/2.<\/p>\n<p>The <em data-effect=\"italics\">direction of intrinsic spin is quantized<\/em>, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the [latex]z[\/latex]-axis, can have only the values<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{S}_{z}={m}_{s}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left({m}_{s}=-\\frac{1}{2},+\\frac{1}{2}\\right)[\/latex]<\/div>\n<p id=\"import-auto-id3047316\">for electrons. [latex]{S}_{z}[\/latex] is the <span data-type=\"term\" id=\"import-auto-id2973845\">[latex]z[\/latex]-component of spin angular momentum<\/span> and [latex]{m}_{s}[\/latex] is the <span data-type=\"term\" id=\"import-auto-id1571054\">spin projection quantum number<\/span>. For electrons, [latex]s[\/latex] can only be 1\/2, and [latex]{m}_{s}[\/latex] can be either +1\/2 or \u20131\/2. Spin projection [latex]{m}_{s}\\text{=+}1\/2[\/latex] is referred to as <em data-effect=\"italics\">spin up<\/em>, whereas [latex]{m}_{s}=-1\/2[\/latex] is called <em data-effect=\"italics\">spin down<\/em>. These are illustrated in <a href=\"\/contents\/fea9523c-93cd-487c-bb4c-b2b367f5c279@2#import-auto-id1870582\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1577628\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Intrinsic Spin<\/div>\n<p>In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles [latex]s[\/latex] is half-integral, whereas for others [latex]s[\/latex] is integral\u2014there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have [latex]s=1\/2[\/latex], whereas photons have [latex]s=1[\/latex], and other particles called pions have [latex]s=0[\/latex], and so on.<\/p>\n<\/div>\n<p id=\"import-auto-id2488547\">To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form [latex]\\left(\\mathrm{n, l,}\\phantom{\\rule{0.25em}{0ex}}{m}_{l},\\phantom{\\rule{0.25em}{0ex}}{m}_{s}\\right)[\/latex] \u2014see <a href=\"#import-auto-id3013526\" class=\"autogenerated-content\">(Figure)<\/a> <em data-effect=\"italics\">For electrons in atoms<\/em>, the principal quantum number can have the values [latex]n=1, 2, 3, ...[\/latex]. Once [latex]n[\/latex] is known, the values of the angular momentum quantum number are limited to [latex]l=1, 2, 3, ...,n-1[\/latex]. For a given value of [latex]l[\/latex], the angular momentum projection quantum number can have only the values [latex]{m}_{l}=-l,\\phantom{\\rule{0.25em}{0ex}}-l\\phantom{\\rule{0.25em}{0ex}}+1, ...,-1, 0, 1, ...,\\phantom{\\rule{0.25em}{0ex}}l-1,\\phantom{\\rule{0.25em}{0ex}}l[\/latex]. Electron spin is independent of [latex]\\mathrm{n, l,}[\/latex] and [latex]{m}_{l}[\/latex], always having [latex]s=1\/2[\/latex]. The spin projection quantum number can have two values, [latex]{m}_{s}=1\/2\\phantom{\\rule{0.25em}{0ex}}\\text{or}\\phantom{\\rule{0.25em}{0ex}}-1\/2[\/latex].<\/p>\n<table id=\"import-auto-id3013526\" summary=\"Three-column table titled Atomic Quantum Numbers. The columns, from left to right, are titled Name, Symbol, and Allowed Values.\">\n<caption><span data-type=\"title\">Atomic Quantum Numbers<\/span><\/caption>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Symbol<\/th>\n<th>Allowed values<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Principal quantum number<\/td>\n<td>[latex]n[\/latex]<\/td>\n<td>[latex]1, 2, 3, ...[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angular momentum<\/td>\n<td>[latex]l[\/latex]<\/td>\n<td>[latex]0, 1, 2, ...n-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angular momentum projection<\/td>\n<td>[latex]{m}_{l}[\/latex]<\/td>\n<td>[latex]-l,\\phantom{\\rule{0.25em}{0ex}}-l\\phantom{\\rule{0.25em}{0ex}}+1, ...,\\phantom{\\rule{0.25em}{0ex}}-1, 0, 1, ...,\\phantom{\\rule{0.25em}{0ex}}l-1,\\phantom{\\rule{0.25em}{0ex}}l\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{or}\\phantom{\\rule{0.25em}{0ex}}0, \u00b11, \u00b12, ...,\\phantom{\\rule{0.25em}{0ex}}\u00b1l\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Spin<a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/td>\n<td>[latex]s[\/latex]<\/td>\n<td>[latex]\\text{1\/2}\\left(\\text{electrons}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Spin projection<\/td>\n<td>[latex]{m}_{s}[\/latex]<\/td>\n<td>[latex]-1\/2,\\phantom{\\rule{0.25em}{0ex}}+1\/2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id3081997\"><a href=\"#import-auto-id1427798\" class=\"autogenerated-content\">(Figure)<\/a> shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements\u2014each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1427798\">\n<div class=\"bc-figcaption figcaption\">Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as [latex]\\left(n,\\phantom{\\rule{0.25em}{0ex}}l,\\phantom{\\rule{0.25em}{0ex}}{m}_{l}\\right)[\/latex]. The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1427799\" data-alt=\"The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_31_08_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.\" width=\"400\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3094028\">We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Stern-Gerlach Experiment<\/div>\n<p id=\"eip-650\">The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1274145\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/c832c456e9538c3a503b8b6ea3963539ea241ea9\/stern-gerlach_en.jar\">Stern-Gerlach Experiment<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_31.8\" data-alt=\"\"><a href=\"\/resources\/c832c456e9538c3a503b8b6ea3963539ea241ea9\/stern-gerlach_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3223976\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3089619\">\n<li id=\"import-auto-id3399326\">Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number [latex]n[\/latex] labels the basic states of a system and is given by\n<div data-type=\"equation\" class=\"equation\">[latex]n=1,\\phantom{\\rule{0.25em}{0ex}}2,\\phantom{\\rule{0.25em}{0ex}}3,\\text{.}\\text{.}\\text{.}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1506968\">The magnitude of angular momentum is given by\n<div data-type=\"equation\" class=\"equation\">[latex]L=\\sqrt{l\\left(l+1\\right)}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left(l=0, 1, 2, ...,\\phantom{\\rule{0.25em}{0ex}}n-1\\right),[\/latex]<\/div>\n<p>where [latex]l[\/latex] is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the [latex]z[\/latex]-axis is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{L}_{z}={m}_{l}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left({m}_{l}=-l,-l+1, ...,\\phantom{\\rule{0.25em}{0ex}}-1, 0, 1, ...\\phantom{\\rule{0.25em}{0ex}}l-1,\\phantom{\\rule{0.25em}{0ex}}l\\right)\\text{,}[\/latex]<\/div>\n<p>where [latex]{L}_{z}[\/latex] is the [latex]z[\/latex]-component of the angular momentum and [latex]{m}_{l}[\/latex] is the angular momentum projection quantum number. Similarly, the electron\u2019s intrinsic spin angular momentum [latex]S[\/latex] is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]S=\\sqrt{s\\left(s+1\\right)}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\text{(}s=1\/2\\phantom{\\rule{0.25em}{0ex}}\\text{for electrons),}[\/latex]<\/div>\n<p>[latex]s[\/latex] is defined to be the spin quantum number. Finally, the direction of the electron\u2019s spin along the [latex]z[\/latex]-axis is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{S}_{z}={m}_{s}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}\\left({m}_{s}=-\\frac{1}{2},+\\frac{1}{2}\\right)\\text{,}[\/latex]<\/div>\n<p>where [latex]{S}_{z}[\/latex] is the [latex]z[\/latex]-component of spin angular momentum and [latex]{m}_{s}[\/latex] is the spin projection quantum number. Spin projection [latex]{m}_{s}\\text{=+}1\/2[\/latex] is referred to as spin up, whereas [latex]{m}_{s}=-1\/2[\/latex] is called spin down. <a href=\"#import-auto-id3013526\" class=\"autogenerated-content\">(Figure)<\/a> summarizes the atomic quantum numbers and their allowed values.\n        <\/p><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2514678\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3259487\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3259490\">\n<p id=\"import-auto-id2453093\">Define the quantum numbers [latex]\\mathrm{n, l,}\\phantom{\\rule{0.25em}{0ex}}{m}_{l}\\mathrm{, s}[\/latex], and [latex]{m}_{s}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1859624\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1859626\">\n<p id=\"import-auto-id3417998\">For a given value of [latex]n[\/latex], what are the allowed values of [latex]l[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3033126\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2017403\">\n<p id=\"import-auto-id2683190\">For a given value of [latex]l[\/latex], what are the allowed values of [latex]{m}_{l}[\/latex]? What are the allowed values of [latex]{m}_{l}[\/latex] for a given value of [latex]n[\/latex]? Give an example in each case.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3224054\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3081607\">\n<p id=\"import-auto-id3080460\">List all the possible values of [latex]s[\/latex] and [latex]{m}_{s}[\/latex] for an electron. Are there particles for which these values are different? The same?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2402788\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1599368\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1599371\">\n<p id=\"import-auto-id3399259\">If an atom has an electron in the [latex]n=5[\/latex] state with [latex]{m}_{l}=3[\/latex], what are the possible values of [latex]l[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1915860\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3250310\">[latex]l=4, 3[\/latex] are possible since [latex]l&lt;n[\/latex] and [latex]\\mid {m}_{l}\\mid \\text{\u2264}l[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3033541\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3033543\">\n<p id=\"import-auto-id2011600\">An atom has an electron with [latex]{m}_{l}=2[\/latex]. What is the smallest value of [latex]n[\/latex] for this electron?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3354859\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1972640\">\n<p id=\"import-auto-id3172796\">What are the possible values of [latex]{m}_{l}[\/latex] for an electron in the [latex]n=4[\/latex] state?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1427934\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id1594122\">[latex]n=4\u21d2l=3, 2, 1, 0\u21d2{m}_{l}=\u00b13,\\phantom{\\rule{0.25em}{0ex}}\u00b1\\text{2,}\\phantom{\\rule{0.25em}{0ex}}\u00b1\\text{1, 0}[\/latex] are possible.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2659412\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2659415\">\n<p id=\"import-auto-id2931417\">What, if any, constraints does a value of [latex]{m}_{l}=1[\/latex] place on the other quantum numbers for an electron in an atom?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2663083\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1921578\">\n<p id=\"import-auto-id3045575\">(a) Calculate the magnitude of the angular momentum for an [latex]l=1[\/latex] electron. (b) Compare your answer to the value Bohr proposed for the [latex]n=1[\/latex] state.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1870060\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3246769\">(a) [latex]1\\text{.}\\text{49}\u00d7{\\text{10}}^{-\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot s[\/latex]<\/p>\n<p id=\"import-auto-id2931025\">(b) [latex]1\\text{.}\\text{06}\u00d7{\\text{10}}^{-\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot s[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2442300\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3063919\">\n<p id=\"import-auto-id2437604\">(a) What is the magnitude of the angular momentum for an [latex]l=1[\/latex] electron? (b) Calculate the magnitude of the electron\u2019s spin angular momentum. (c) What is the ratio of these angular momenta?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id957081\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3154047\">\n<p id=\"import-auto-id3258505\">Repeat <a href=\"#fs-id2442300\" class=\"autogenerated-content\">(Figure)<\/a> for [latex]l=3[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1575979\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3356255\">(a) [latex]3\\text{.}\\text{66}\u00d7{\\text{10}}^{-\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot s[\/latex]<\/p>\n<p id=\"import-auto-id1449529\">(b) [latex]s=9\\text{.}\\text{13}\u00d7{\\text{10}}^{-\\text{35}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot s[\/latex]<\/p>\n<p id=\"import-auto-id2684257\">(c) [latex]\\frac{L}{S}=\\frac{\\sqrt{\\text{12}}}{\\sqrt{3\/4}}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449672\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3449675\">\n<p id=\"import-auto-id2000896\">(a) How many angles can [latex]L[\/latex] make with the [latex]z[\/latex]-axis for an [latex]l=2[\/latex] electron? (b) Calculate the value of the smallest angle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2991428\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2991431\">\n<p id=\"import-auto-id3423236\">What angles can the spin [latex]S[\/latex] of an electron make with the [latex]z[\/latex]-axis?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2973223\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2488510\">[latex]\\theta =\\text{54.7\u00ba, 125.3\u00ba}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> The spin quantum number <em data-effect=\"italics\">s<\/em> is usually not stated, since it is always 1\/2 for electrons<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2423496\">\n<dt>quantum numbers<\/dt>\n<dd id=\"fs-id3103481\">the values of quantized entities, such as energy and angular momentum<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2423498\">\n<dt>angular momentum quantum number<\/dt>\n<dd id=\"fs-id1916440\">a quantum number associated with the angular momentum of electrons<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2423501\">\n<dt>spin quantum number<\/dt>\n<dd id=\"fs-id2655822\">the quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2236094\">\n<dt>spin projection quantum number<\/dt>\n<dd id=\"fs-id1498878\">quantum number that can be used to calculate the intrinsic electron angular momentum along the  [latex]z[\/latex]-axis<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1498882\">\n<dt>z-component of spin angular momentum<\/dt>\n<dd id=\"fs-id2384934\">component of intrinsic electron spin along the  [latex]z[\/latex]-axis<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2384937\">\n<dt>magnitude of the intrinsic (internal) spin angular momentum<\/dt>\n<dd id=\"fs-id2384941\">given by [latex]S=\\sqrt{s\\left(s+1\\right)}\\frac{h}{2\\pi }\\phantom{\\rule{1.00em}{0ex}}[\/latex] <\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1589940\">\n<dt>z-component of the angular momentum<\/dt>\n<dd id=\"fs-id1589948\">component of orbital angular momentum of electron along the  [latex]z[\/latex]-axis<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define quantum number.<\/li>\n<li>Calculate angle of angular momentum vector with an axis.<\/li>\n<li>Define spin quantum number.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1276239\">Physical characteristics that are quantized\u2014such as energy, charge, and angular momentum\u2014are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of <span data-type=\"term\">quantum numbers<\/span>, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules\u2014all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.<\/p>\n<p id=\"import-auto-id2625656\">The <em data-effect=\"italics\">energy states of bound systems are quantized<\/em>, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cd4d6e7b054802a20b70a1f153a8ffc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#110;&#125;&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;&#49;&#47;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e5dd2b41cddb9382a4ec010fa3f42c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/>. We define<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> to be the principal quantum number that labels the basic states of a system. The lowest-energy state has<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a51661019cc26a5931f8cb0d5fd63f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>, the first excited state has<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8b4c6cd9d27ba344abe355a47d378bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/>, and so on. Thus the allowed values for the principal quantum number are<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d5790dec1e34163fd66fe9058eea9ddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#46;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1390187\">This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, as can other characteristics (such as the orbital radii of the hydrogen atom).<\/p>\n<p id=\"import-auto-id3063639\">The fact that the <em data-effect=\"italics\">magnitude of angular momentum is quantized<\/em> was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> can have only the values<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea965de327b57aacec94a1dbb2f0dfe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#61;&#48;&#44;&#32;&#49;&#44;&#32;&#50;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"320\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3069597\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> is defined to be the <span data-type=\"term\" id=\"import-auto-id1432038\">angular momentum quantum number<\/span>. The rule for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> in atoms is given in the parentheses. Given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> can be any integer from zero up to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f30b71e7fcec69d119f30f67cf09c975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/>. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d48be04aadb63e5661f86d0948d7553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> can be 0, 1, 2, or 3.<\/p>\n<p>Note that for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a51661019cc26a5931f8cb0d5fd63f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3516542a5278fdde3c8da9c28e1520cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#47;&#50;&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -5px;\" \/> as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in <a href=\"\/contents\/aedd88f0-3f74-4137-b2f1-67af8b4d7832@3#import-auto-id2396594\" class=\"autogenerated-content\">(Figure)<\/a>. The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg\u2019s uncertainty principle\u2014the electron\u2019s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8b4c6cd9d27ba344abe355a47d378bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/>, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> can be either 0 or 1, according to the rule in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c9da164acd544f6852c0c114857532b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/> . Similarly, for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-85761a607075b960ff00638d721cfe9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> can be 0, 1, or 2. It is often most convenient to state the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>, a simple integer, rather than calculating the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c9da164acd544f6852c0c114857532b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/>. For example, for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab7c30f3fbbd6b506c3447fd8dbbd2c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\" \/>, we see that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6fa9d333823ed49e42e02c7c7dac6168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#57;&#48;&#125;&#104;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#115;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"415\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>It is much simpler to state <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab7c30f3fbbd6b506c3447fd8dbbd2c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id2992778\">As recognized in the Zeeman effect, the <em data-effect=\"italics\">direction of angular momentum is quantized<\/em>. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis, can have only certain values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd4e8eec0ff01ff139749fd6499db84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/>. The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd4e8eec0ff01ff139749fd6499db84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/><em data-effect=\"italics\"> are<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e8175f9cff8f66a1c8a7be13745be62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#45;&#108;&#44;&#45;&#108;&#43;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#44;&#32;&#48;&#44;&#32;&#49;&#44;&#32;&#46;&#46;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#45;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"426\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd4e8eec0ff01ff139749fd6499db84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id2953765\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-component of the angular momentum<\/span> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> is the angular momentum projection quantum number. The rule in parentheses for the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> is that it can range from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cee1d4ddc1db3b3f4a71439b73ec5e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> in steps of one. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab7c30f3fbbd6b506c3447fd8dbbd2c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> can have the five values \u20132, \u20131, 0, 1, and 2. Each <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1996787\">\n<div class=\"bc-figcaption figcaption\">The component of a given angular momentum along the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a20249cd06149045d2aaca6012bd789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -1px;\" \/>, for which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-12d59da24628f87e817fdd2ebc29d0e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#45;&#49;&#44;&#32;&#48;&#44;&#32;&#97;&#110;&#100;&#32;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"150\" style=\"vertical-align: -4px;\" \/>. The direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is quantized in the sense that it can have only certain angles relative to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis.<\/div>\n<p><span data-type=\"media\" data-alt=\"The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_31_08_00a.jpg\" data-media-type=\"image\/jpg\" alt=\"The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1419670\">\n<div data-type=\"title\" class=\"title\">What Are the Allowed Directions?<\/div>\n<p id=\"import-auto-id2928454\">Calculate the angles that the angular momentum vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-228b8813a04bf734db1c743abd4c8ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\" \/><strong data-effect=\"bold\"> can make with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a20249cd06149045d2aaca6012bd789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -1px;\" \/>, as illustrated in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>.<\/strong><\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1350660\"><a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a> represents the vectors  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-228b8813a04bf734db1c743abd4c8ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc9d4401717314c045657bac24f41cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\" \/> as usual, with arrows proportional to their magnitudes and pointing in the correct directions. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-228b8813a04bf734db1c743abd4c8ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\" \/><strong data-effect=\"bold\"> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc9d4401717314c045657bac24f41cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\" \/> form a right triangle, with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d66ccc495647a0feaeda36aff5fdafeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#76;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/> being the hypotenuse and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc9d4401717314c045657bac24f41cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\" \/> the adjacent side. This means that the ratio of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-61fa98c09c1a8e2a2d8f34b6b96bc108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#76;&#125;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -3px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d66ccc495647a0feaeda36aff5fdafeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#76;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/> is the cosine of the angle of interest. We can find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-228b8813a04bf734db1c743abd4c8ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc9d4401717314c045657bac24f41cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\" \/> using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c9da164acd544f6852c0c114857532b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c112a1159a6efe69dee9289b6b06baa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#61;&#109;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"77\" style=\"vertical-align: -6px;\" \/>.<\/strong><\/p>\n<p id=\"import-auto-id3093903\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1602621\">We are given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a20249cd06149045d2aaca6012bd789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -1px;\" \/>, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> can be +1, 0, or \u22121. Thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> has the value given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c9da164acd544f6852c0c114857532b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eaa886742dc8f1737c625c7e7461b750_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"152\" style=\"vertical-align: -6px;\" \/><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd4e8eec0ff01ff139749fd6499db84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/><em data-effect=\"italics\"> can have three values, given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9321ac29d68730a2faf7063c39501463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/>.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-250\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8fb1b7c486d26453bfbba46597e2972_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#44;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#38;&#32;&#61;&#38;&#32;&#43;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#38;&#32;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#38;&#32;&#61;&#38;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#92;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#44;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#38;&#32;&#61;&#38;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"280\" style=\"vertical-align: -29px;\" \/><\/div>\n<p id=\"import-auto-id1349581\">As can be seen in <a href=\"#import-auto-id1996787\" class=\"autogenerated-content\">(Figure)<\/a>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3565552c1faf0b2893110da73783f4d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#111;&#115;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#125;&#95;&#123;&#122;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#76;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><br \/>\n and so for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b285ddf79eaec93f4209e8d75aeccfcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/>, we have<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-27b70b86f21a8b8c214499064ba2a170_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#76;&#125;&#95;&#123;&#90;&#125;&#125;&#123;&#76;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#55;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"259\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"import-auto-id2437142\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-380a85af4ef1f9fa00f76a5c0649fb7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#55;&#48;&#55;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2697885\">Similarly, for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e81f4da837ac30da0c99cc20ffc418c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -3px;\" \/>, we find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3e66bd1c9db4be5c9dbbeb6ee0d1111b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"77\" style=\"vertical-align: -3px;\" \/>; thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9ac844832744bdbde423039b11733f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#48;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2655556\">And for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0fcdc79267ccda1a5c50d81706d5529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/>,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2dfb20f7ea7623ae128d4cdeec087495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#76;&#125;&#95;&#123;&#90;&#125;&#125;&#123;&#76;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#61;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#55;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"288\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"import-auto-id2583456\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-20cad128bd5fa14008097a2fa9ca76a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#51;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"221\" style=\"vertical-align: -4px;\" \/><\/div>\n<p><strong>Discussion<\/strong><\/p>\n<p>The angles are consistent with the figure. Only the angle relative to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis is quantized. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> can point in any direction as long as it makes the proper angle with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d69c33b7f07e6d88fef24a6ce1bec793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/> in the example) is for the maximum value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e81f4da837ac30da0c99cc20ffc418c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -3px;\" \/>, namely <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-29fb55bfba6fd5c8501a2e46196d1e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -3px;\" \/>. For that smallest angle,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-509cc860f424d7ce916a0be4775c8474_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#125;&#123;&#76;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#108;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"164\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id1198202\">which approaches 1 as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> becomes very large. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bd4730d50e3c1bed86f0cac04a30f124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: -1px;\" \/>, then <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b575a66b78fa57c473a63a12c9e72f1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/>. Furthermore, for large <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>, there are many values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/>, so that all angles become possible as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> gets very large.<\/em><\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1448501\">\n<h1 data-type=\"title\">Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction<\/h1>\n<p id=\"import-auto-id3421186\">There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have <em data-effect=\"italics\">intrinsic spin<\/em>, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the <span data-type=\"term\" id=\"import-auto-id2436972\">magnitude of the intrinsic (internal) spin angular momentum<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, of an electron is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0a09982531541012684eaa58e9405753_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#125;&#115;&#61;&#49;&#47;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#101;&#108;&#101;&#99;&#116;&#114;&#111;&#110;&#115;&#41;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"331\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is defined to be the <span data-type=\"term\" id=\"import-auto-id1871891\">spin quantum number<\/span>. This is very similar to the quantization of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> given in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c9da164acd544f6852c0c114857532b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/>, except that the only value allowed for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> for electrons is 1\/2.<\/p>\n<p>The <em data-effect=\"italics\">direction of intrinsic spin is quantized<\/em>, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis, can have only the values<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3327bffa133cf615a8f52a2d341357ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"222\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id3047316\">for electrons. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-be0bd1376abba0a46299a64d803cf25f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id2973845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-component of spin angular momentum<\/span> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id1571054\">spin projection quantum number<\/span>. For electrons, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> can only be 1\/2, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/> can be either +1\/2 or \u20131\/2. Spin projection <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-569bb294b52090690ad870acdff01de9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#43;&#125;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -5px;\" \/> is referred to as <em data-effect=\"italics\">spin up<\/em>, whereas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-12577a814e3a6e5106f7109d0bd242f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#61;&#45;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -5px;\" \/> is called <em data-effect=\"italics\">spin down<\/em>. These are illustrated in <a href=\"\/contents\/fea9523c-93cd-487c-bb4c-b2b367f5c279@2#import-auto-id1870582\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1577628\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Intrinsic Spin<\/div>\n<p>In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is half-integral, whereas for others <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is integral\u2014there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-318481f75a4057479e52d2f52aa6a787_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -5px;\" \/>, whereas photons have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c81f9a68260bcbde925109ef89e90fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/>, and other particles called pions have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-49f6fbc1f86c3bbbcdd50374b3d67190_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/>, and so on.<\/p>\n<\/div>\n<p id=\"import-auto-id2488547\">To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e083cc4a9b3dfc804602111fb65dd91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#110;&#44;&#32;&#108;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/> \u2014see <a href=\"#import-auto-id3013526\" class=\"autogenerated-content\">(Figure)<\/a> <em data-effect=\"italics\">For electrons in atoms<\/em>, the principal quantum number can have the values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e5dd2b41cddb9382a4ec010fa3f42c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/>. Once <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is known, the values of the angular momentum quantum number are limited to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c2c044cffdc565afe010ba0087d406a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#46;&#46;&#46;&#44;&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"142\" style=\"vertical-align: -4px;\" \/>. For a given value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>, the angular momentum projection quantum number can have only the values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f71b7b0b62f09c67235125f04a60a6c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#45;&#108;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#108;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#45;&#49;&#44;&#32;&#48;&#44;&#32;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#45;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"313\" style=\"vertical-align: -4px;\" \/>. Electron spin is independent of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-37946dbcc8b65f156c4e0159c167038c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#110;&#44;&#32;&#108;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"27\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/>, always having <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-318481f75a4057479e52d2f52aa6a787_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -5px;\" \/>. The spin projection quantum number can have two values, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb1543027521cd4c472b10024109fa5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#61;&#49;&#47;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<table id=\"import-auto-id3013526\" summary=\"Three-column table titled Atomic Quantum Numbers. The columns, from left to right, are titled Name, Symbol, and Allowed Values.\">\n<caption><span data-type=\"title\">Atomic Quantum Numbers<\/span><\/caption>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Symbol<\/th>\n<th>Allowed values<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Principal quantum number<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ce8aa0fed08e8351f2ee5aa7a9e922_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Angular momentum<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18047c5aaa7fa0de357a5e25ed780a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#44;&#32;&#49;&#44;&#32;&#50;&#44;&#32;&#46;&#46;&#46;&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Angular momentum projection<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ebf29287b2d5b5544f590197bfc9e997_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#108;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#108;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#44;&#32;&#48;&#44;&#32;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#45;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#44;&#32;&plusmn;&#49;&#44;&#32;&plusmn;&#50;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&plusmn;&#108;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"405\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Spin<a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6877f9c1f5bdc30bcc93d910525e21d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#47;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#116;&#114;&#111;&#110;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Spin projection<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8a68417c430e88edac384f942237d57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#47;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id3081997\"><a href=\"#import-auto-id1427798\" class=\"autogenerated-content\">(Figure)<\/a> shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements\u2014each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1427798\">\n<div class=\"bc-figcaption figcaption\">Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8e3206e067eba09a4eff915d5fe15020_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>. The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1427799\" data-alt=\"The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_31_08_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3094028\">We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Stern-Gerlach Experiment<\/div>\n<p id=\"eip-650\">The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1274145\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/c832c456e9538c3a503b8b6ea3963539ea241ea9\/stern-gerlach_en.jar\">Stern-Gerlach Experiment<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_31.8\" data-alt=\"\"><a href=\"\/resources\/c832c456e9538c3a503b8b6ea3963539ea241ea9\/stern-gerlach_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3223976\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3089619\">\n<li id=\"import-auto-id3399326\">Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> labels the basic states of a system and is given by\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-568d6867e56ffb55353b83d9582b2e13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1506968\">The magnitude of angular momentum is given by\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-07cfb90b2e4936aa45953ff14b68112d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#108;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#108;&#61;&#48;&#44;&#32;&#49;&#44;&#32;&#50;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"320\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/> is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e8175f9cff8f66a1c8a7be13745be62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#45;&#108;&#44;&#45;&#108;&#43;&#49;&#44;&#32;&#46;&#46;&#46;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#44;&#32;&#48;&#44;&#32;&#49;&#44;&#32;&#46;&#46;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#45;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#108;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"426\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd4e8eec0ff01ff139749fd6499db84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-component of the angular momentum and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> is the angular momentum projection quantum number. Similarly, the electron\u2019s intrinsic spin angular momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0a09982531541012684eaa58e9405753_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#125;&#115;&#61;&#49;&#47;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#101;&#108;&#101;&#99;&#116;&#114;&#111;&#110;&#115;&#41;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"331\" style=\"vertical-align: -6px;\" \/><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is defined to be the spin quantum number. Finally, the direction of the electron\u2019s spin along the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-23081619a9873d1db2bc7bdf17b1603c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#122;&#125;&#61;&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"231\" style=\"vertical-align: -7px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-be0bd1376abba0a46299a64d803cf25f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-component of spin angular momentum and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/> is the spin projection quantum number. Spin projection <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-569bb294b52090690ad870acdff01de9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#43;&#125;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -5px;\" \/> is referred to as spin up, whereas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-12577a814e3a6e5106f7109d0bd242f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;&#61;&#45;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -5px;\" \/> is called spin down. <a href=\"#import-auto-id3013526\" class=\"autogenerated-content\">(Figure)<\/a> summarizes the atomic quantum numbers and their allowed values.\n        <\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2514678\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3259487\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3259490\">\n<p id=\"import-auto-id2453093\">Define the quantum numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dcbe5f96f52637ea9cd3236061a423d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#110;&#44;&#32;&#108;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#44;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"68\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1859624\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1859626\">\n<p id=\"import-auto-id3417998\">For a given value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, what are the allowed values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3033126\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2017403\">\n<p id=\"import-auto-id2683190\">For a given value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>, what are the allowed values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/>? What are the allowed values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> for a given value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>? Give an example in each case.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3224054\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3081607\">\n<p id=\"import-auto-id3080460\">List all the possible values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d2d5cd720dde70463c6cd7244570a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\" \/> for an electron. Are there particles for which these values are different? The same?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2402788\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1599368\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1599371\">\n<p id=\"import-auto-id3399259\">If an atom has an electron in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6f852454e2c6c16d005bca006d38cbc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> state with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e7bedc2eb3a1798a9da3cd7cf3a149e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -3px;\" \/>, what are the possible values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-502276c66966e5a861539c7de60c26c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"5\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1915860\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3250310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-da6bc733cda5334ea9831ad47a102eff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#52;&#44;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/> are possible since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c949936edd362ff33aa72af60d47895e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#60;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f37ef500ee2073a0ad79b2474cbdece8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#105;&#100;&#32;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#92;&#109;&#105;&#100;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#108;&#101;&#125;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3033541\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3033543\">\n<p id=\"import-auto-id2011600\">An atom has an electron with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eedd824d23ab55e17c68580f08e0ba78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\" \/>. What is the smallest value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> for this electron?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3354859\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1972640\">\n<p id=\"import-auto-id3172796\">What are the possible values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76732a572f6c9fdce08b5d74dd7aced4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"20\" style=\"vertical-align: -3px;\" \/> for an electron in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d48be04aadb63e5661f86d0948d7553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> state?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1427934\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id1594122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2896813b90ad3165645cee3d74d097c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#52;&#8658;&#108;&#61;&#51;&#44;&#32;&#50;&#44;&#32;&#49;&#44;&#32;&#48;&#8658;&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&plusmn;&#51;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&plusmn;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&plusmn;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#44;&#32;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"244\" style=\"vertical-align: -4px;\" \/> are possible.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2659412\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2659415\">\n<p id=\"import-auto-id2931417\">What, if any, constraints does a value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c8c49e3152a6a4b1872a57518743b3ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#95;&#123;&#108;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\" \/> place on the other quantum numbers for an electron in an atom?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2663083\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1921578\">\n<p id=\"import-auto-id3045575\">(a) Calculate the magnitude of the angular momentum for an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a20249cd06149045d2aaca6012bd789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -1px;\" \/> electron. (b) Compare your answer to the value Bohr proposed for the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a51661019cc26a5931f8cb0d5fd63f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> state.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1870060\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3246769\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4adfb79da1aa2b6c26db4b62047b19f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2931025\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c689a0f593d54785a729fabc4b226ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2442300\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3063919\">\n<p id=\"import-auto-id2437604\">(a) What is the magnitude of the angular momentum for an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a20249cd06149045d2aaca6012bd789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -1px;\" \/> electron? (b) Calculate the magnitude of the electron\u2019s spin angular momentum. (c) What is the ratio of these angular momenta?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id957081\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3154047\">\n<p id=\"import-auto-id3258505\">Repeat <a href=\"#fs-id2442300\" class=\"autogenerated-content\">(Figure)<\/a> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ec8d721b0787eef7de79aae4e85381bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1575979\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3356255\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ec585ce9ce30904d579f501737236316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id1449529\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4ebb26dd4fa7a9318e13dc606a912c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"141\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2684257\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60bcdc95e96eb224dd58909e1da25cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#83;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#47;&#52;&#125;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"107\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449672\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3449675\">\n<p id=\"import-auto-id2000896\">(a) How many angles can <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> make with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis for an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab7c30f3fbbd6b506c3447fd8dbbd2c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\" \/> electron? (b) Calculate the value of the smallest angle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2991428\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2991431\">\n<p id=\"import-auto-id3423236\">What angles can the spin <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> of an electron make with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2973223\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2488510\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f837fdcc258e303c05291abb8452b1ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#52;&#46;&#55;&ordm;&#44;&#32;&#49;&#50;&#53;&#46;&#51;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> The spin quantum number <em data-effect=\"italics\">s<\/em> is usually not stated, since it is always 1\/2 for electrons<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2423496\">\n<dt>quantum numbers<\/dt>\n<dd id=\"fs-id3103481\">the values of quantized entities, such as energy and angular momentum<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2423498\">\n<dt>angular momentum quantum number<\/dt>\n<dd id=\"fs-id1916440\">a quantum number associated with the angular momentum of electrons<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2423501\">\n<dt>spin quantum number<\/dt>\n<dd id=\"fs-id2655822\">the quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2236094\">\n<dt>spin projection quantum number<\/dt>\n<dd id=\"fs-id1498878\">quantum number that can be used to calculate the intrinsic electron angular momentum along the  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1498882\">\n<dt>z-component of spin angular momentum<\/dt>\n<dd id=\"fs-id2384934\">component of intrinsic electron spin along the  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>-axis<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2384937\">\n<dt>magnitude of the intrinsic (internal) spin angular momentum<\/dt>\n<dd id=\"fs-id2384941\">given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb2458e61cff81bf33284a86c776b6b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#104;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -6px;\" \/> <\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1589940\">\n<dt>z-component of the angular momentum<\/dt>\n<dd id=\"fs-id1589948\">component of orbital angular momentum of electron along the  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" 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