{"id":195,"date":"2019-04-09T00:42:35","date_gmt":"2019-04-09T04:42:35","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&#038;p=195"},"modified":"2019-04-12T18:57:36","modified_gmt":"2019-04-12T22:57:36","slug":"3-3-the-schr%d3%a7dinger-equation","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/3-3-the-schr%d3%a7dinger-equation\/","title":{"raw":"3.3 The Schr\u04e7dinger Equation","rendered":"3.3 The Schr\u04e7dinger Equation"},"content":{"raw":"<div data-type=\"abstract\" id=\"3711\" class=\"ui-has-child-title\"><header>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Describe the role Schr\u04e7dinger\u2019s equation plays in quantum mechanics<\/li>\r\n \t<li>Explain the difference between time-dependent and -independent Schr\u04e7dinger\u2019s equations<\/li>\r\n \t<li>Interpret the solutions of Schr\u04e7dinger\u2019s equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In the preceding two sections, we described how to use a quantum mechanical wave function and discussed Heisenberg\u2019s uncertainty principle. In this section, we present a complete and formal theory of quantum mechanics that can be used to make predictions. In developing this theory, it is helpful to review the wave theory of light. For a light wave, the electric field<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">,<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">) obeys the relation<\/span>\r\n\r\n<\/header><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170903853347\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-156-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-156-Frame\"><span class=\"MathJax_MathContainer\"><span>\u22022E\u2202x2=1c2\u22022E\u2202t2,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.17]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">c<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the speed of light and the symbol<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-157-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u2202<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">represents a<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">partial derivative<\/em><span style=\"font-size: 14pt\">. (Recall from<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:4b45e3be-3d21-4d22-bdff-e6d0fe4d79f0\" data-page=\"1\" style=\"font-size: 14pt\">Oscillations<\/a><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">that a partial derivative is closely related to an ordinary derivative, but involves functions of more than one variable. When taking the partial derivative of a function by a certain variable, all other variables are held constant.) A light wave consists of a very large number of photons, so the quantity<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-158-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|E(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">can interpreted as a probability density of finding a single photon at a particular point in space (for example, on a viewing screen).<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902194849\">There are many solutions to this equation. One solution of particular importance is<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902118559\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-159-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-159-Frame\"><span class=\"MathJax_MathContainer\"><span>E(x,t)=Asin(kx\u2212\u03c9t),<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.18]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the amplitude of the electric field,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">k<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the wave number, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-160-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c9<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the angular frequency. Combing this equation with<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.17<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902186856\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-161-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-161-Frame\"><span class=\"MathJax_MathContainer\"><span>k2=\u03c92c2.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.19]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">According to de Broglie\u2019s equations, we have<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-162-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">p=\u210fk<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-163-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E=\u210f\u03c9<\/span><\/span><span style=\"font-size: 14pt\">. Substituting these equations in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.19<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170904152351\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-164-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-164-Frame\"><span class=\"MathJax_MathContainer\"><span>p=Ec,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.20]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">or<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170904252568\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-165-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-165-Frame\"><span class=\"MathJax_MathContainer\"><span>E=pc.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.21]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Therefore, according to Einstein\u2019s general energy-momentum equation (Equation 1.11),<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.17<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">describes a particle with a zero rest mass. This is consistent with our knowledge of a photon.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170904153453\">This process can be reversed. We can begin with the energy-momentum equation of a particle and then ask what wave equation corresponds to it. The energy-momentum equation of a nonrelativistic particle in one dimension is<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170903900466\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-166-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-166-Frame\"><span class=\"MathJax_MathContainer\"><span>E=p22m+U(x,t),<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.22]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">p<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the momentum,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">m<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the mass, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">U<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the potential energy of the particle. The wave equation that goes with it turns out to be a key equation in quantum mechanics, called<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term301\" style=\"font-size: 14pt\">Schr\u04e7dinger\u2019s time-dependent equation<\/span><span style=\"font-size: 14pt\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1170903798652\" class=\"ui-has-child-title\"><header>\r\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\" id=\"13370\">THE SCHR\u04e6DINGER TIME-DEPENDENT EQUATION<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1170904065074\">The equation describing the energy and momentum of a wave function is known as the Schr\u04e7dinger equation:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902110094\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-167-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-167-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22m\u22022\u03a8(x,t)\u2202x2+U(x,t)\u03a8(x,t)=i\u210f\u2202\u03a8(x,t)\u2202t.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.23]<\/span><\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1170902089474\">As described in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:eb80e8be-e956-45ae-b17e-ea0c66d4bbb8\" data-page=\"1\">Potential Energy and Conservation of Energy<\/a>, the force on the particle described by this equation is given by<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902179935\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-168-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-168-Frame\"><span class=\"MathJax_MathContainer\"><span>F=\u2212\u2202U(x,t)\u2202x.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.24]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">This equation plays a role in quantum mechanics similar to Newton\u2019s second law in classical mechanics. Once the potential energy of a particle is specified\u2014or, equivalently, once the force on the particle is specified\u2014we can solve this differential equation for the wave function. The solution to Newton\u2019s second law equation (also a differential equation) in one dimension is a function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">) that specifies where an object is at any time<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">. The solution to Schr\u04e7dinger\u2019s time-dependent equation provides a tool\u2014the wave function\u2014that can be used to determine where the particle is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">likely<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to be. This equation can be also written in two or three dimensions. Solving Schr\u04e7dinger\u2019s time-dependent equation often requires the aid of a computer.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902061967\">Consider the special case of a free particle. A free particle experiences no force<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-169-Frame\"><span class=\"MathJax_MathContainer\"><span>(F=0).<\/span><\/span><\/span><span>\u00a0<\/span>Based on<span>\u00a0<\/span>Equation 3.24, this requires only that<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902195526\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-170-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-170-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x,t)=U0=constant.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.25]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">For simplicity, we set<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-171-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0=0<\/span><\/span><span style=\"font-size: 14pt\">. Schr\u04e7dinger\u2019s equation then reduces to<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170904052586\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-172-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-172-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22m\u22022\u03a8(x,t)\u2202x2=i\u210f\u2202\u03a8(x,t)\u2202t.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.26]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">A valid solution to this equation is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902196880\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-173-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-173-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Aei(kx\u2212\u03c9t).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.27]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Not surprisingly, this solution contains an<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term302\" style=\"font-size: 14pt\">imaginary number<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-174-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">(i=\u22121)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">because the differential equation itself contains an imaginary number. As stressed before, however, quantum-mechanical predictions depend only on<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-175-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03a8(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">, which yields completely real values. Notice that the real plane-wave solutions,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-176-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=Asin(kx\u2212\u03c9t)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-177-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=Acos(kx\u2212\u03c9t),<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">do not obey Schr\u00f6dinger\u2019s equation. The temptation to think that a wave function can be seen, touched, and felt in nature is eliminated by the appearance of an imaginary number. In Schr\u04e7dinger\u2019s theory of quantum mechanics, the wave function is merely a tool for calculating things.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902183058\">If the potential energy function (<em data-effect=\"italics\">U<\/em>) does not depend on time, it is possible to show that<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-178-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=\u03c8(x)e\u2212i\u03c9t<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.28]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902089128\">satisfies Schr\u04e7dinger\u2019s time-dependent equation, where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-179-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>is a<span>\u00a0<\/span><em data-effect=\"italics\">time<\/em>-independent function and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-180-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u2212i\u03c9t<\/span><\/span><\/span><span>\u00a0<\/span>is a<span>\u00a0<\/span><em data-effect=\"italics\">space<\/em>-independent function. In other words, the wave function is<span>\u00a0<\/span><em data-effect=\"italics\">separable<\/em><span>\u00a0<\/span>into two parts: a space-only part and a time-only part. The factor<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-181-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u2212i\u03c9t<\/span><\/span><\/span><span>\u00a0<\/span>is sometimes referred to as a<span>\u00a0<\/span><span data-type=\"term\" id=\"term303\">time-modulation factor<\/span><span>\u00a0<\/span>since it modifies the space-only function. According to de Broglie, the energy of a matter wave is given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-182-Frame\"><span class=\"MathJax_MathContainer\"><span>E=\u210f\u03c9<\/span><\/span><\/span>, where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is its total energy. Thus, the above equation can also be written as<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170904230883\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-183-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-183-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=\u03c8(x)e\u2212iEt\/\u210f.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.29]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Any linear combination of such states (mixed state of energy or momentum) is also valid solution to this equation. Such states can, for example, describe a localized particle (see<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.9)<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.5<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header><span style=\"font-size: 1rem\">A particle with mass<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">m<\/em><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">is moving along the<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"font-size: 1rem\">-axis in a potential given by the potential energy function<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-184-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">U(x)=0.5m\u03c92x2<\/span><\/span><span style=\"font-size: 1rem\">. Compute the product<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-185-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)*U(x)\u03a8(x,t).<\/span><\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">Express your answer in terms of the time-independent wave function,<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-186-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c8(x).<\/span><\/span><\/header><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Combining<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.23<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.28, Schr\u00f6dinger\u2019s time-dependent equation reduces to<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-187-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8(x)dx2+U(x)\u03c8(x)=E\u03c8(x),<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.30]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170904202033\">where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the total energy of the particle (a real number). This equation is called<span>\u00a0<\/span><span data-type=\"term\" id=\"term304\">Schr\u04e7dinger\u2019s time-independent equation<\/span>. Notice that we use \u201cbig psi\u201d<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-188-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03a8)<\/span><\/span><\/span><span>\u00a0<\/span>for the time-dependent wave function and \u201clittle psi\u201d<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-189-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03c8)<\/span><\/span><\/span><span>\u00a0<\/span>for the time-independent wave function. The wave-function solution to this equation must be multiplied by the time-modulation factor to obtain the time-dependent wave function.<\/p>\r\n<p id=\"fs-id1170904272107\">In the next sections, we solve Schr\u04e7dinger\u2019s time-independent equation for three cases: a quantum particle in a box, a simple harmonic oscillator, and a quantum barrier. These cases provide important lessons that can be used to solve more complicated systems. The time-independent wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-190-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>solutions must satisfy three conditions:<\/p>\r\n\r\n<ul id=\"fs-id1170902363308\" data-bullet-style=\"bullet\">\r\n \t<li><span class=\"MathJax_MathML\" id=\"MathJax-Element-191-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>must be a continuous function.<\/li>\r\n \t<li>The first derivative of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-192-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>with respect to space,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-193-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8(x)\/dx<\/span><\/span><\/span>, must be continuous, unless<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-194-Frame\"><span class=\"MathJax_MathContainer\"><span>V(x)=\u221e<\/span><\/span><\/span>.<\/li>\r\n \t<li><span class=\"MathJax_MathML\" id=\"MathJax-Element-195-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>must not diverge (\u201cblow up\u201d) at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-196-Frame\"><span class=\"MathJax_MathContainer\"><span>x=\u00b1\u221e.<\/span><\/span><\/span><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170902216092\">The first condition avoids sudden jumps or gaps in the wave function. The second condition requires the wave function to be smooth at all points, except in special cases. (In a more advanced course on quantum mechanics, for example, potential spikes of infinite depth and height are used to model solids). The third condition requires the wave function be normalizable. This third condition follows from Born\u2019s interpretation of quantum mechanics. It ensures that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-197-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8(x)|2<\/span><\/span><\/span><span>\u00a0<\/span>is a finite number so we can use it to calculate probabilities.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.6<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header>\r\n<div class=\"os-title\"><span style=\"font-size: 1rem\">Which of the following wave functions is a valid wave-function solution for Schr\u04e7dinger\u2019s equation?<\/span><\/div>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div class=\"os-hasSolution unnumbered\" data-type=\"exercise\" id=\"fs-id1170902214402\"><section>\r\n<div data-type=\"problem\" id=\"fs-id1170903855855\">\r\n<div class=\"os-problem-container\">\r\n\r\n<span data-alt=\"Three graphs of Psi of x versus x are shown. The first rises then drops discontinuously to a lower value, rises again and then has a constant value. The second function looks like a breaking wave, with a crest overtaking the base. The third increases exponentially to infinity.\" data-type=\"media\" id=\"fs-id1170904111214\"><img alt=\"Three graphs of Psi of x versus x are shown. The first rises then drops discontinuously to a lower value, rises again and then has a constant value. The second function looks like a breaking wave, with a crest overtaking the base. The third increases exponentially to infinity.\" data-media-type=\"image\/jpeg\" id=\"5457\" src=\"https:\/\/cnx.org\/resources\/a98eb833bef078ae0a706e5084d8e2649442cc72\" class=\"alignnone\" width=\"649\" height=\"154\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>","rendered":"<div data-type=\"abstract\" id=\"3711\" class=\"ui-has-child-title\">\n<header>\n<div class=\"textbox textbox--learning-objectives\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe the role Schr\u04e7dinger\u2019s equation plays in quantum mechanics<\/li>\n<li>Explain the difference between time-dependent and -independent Schr\u04e7dinger\u2019s equations<\/li>\n<li>Interpret the solutions of Schr\u04e7dinger\u2019s equation<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In the preceding two sections, we described how to use a quantum mechanical wave function and discussed Heisenberg\u2019s uncertainty principle. In this section, we present a complete and formal theory of quantum mechanics that can be used to make predictions. In developing this theory, it is helpful to review the wave theory of light. For a light wave, the electric field<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">,<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">) obeys the relation<\/span><\/p>\n<div data-type=\"equation\" id=\"fs-id1170903853347\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-156-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-156-Frame\"><span class=\"MathJax_MathContainer\"><span>\u22022E\u2202x2=1c2\u22022E\u2202t2,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.17]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">c<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the speed of light and the symbol<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-157-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u2202<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">represents a<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">partial derivative<\/em><span style=\"font-size: 14pt\">. (Recall from<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:4b45e3be-3d21-4d22-bdff-e6d0fe4d79f0\" data-page=\"1\" style=\"font-size: 14pt\">Oscillations<\/a><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">that a partial derivative is closely related to an ordinary derivative, but involves functions of more than one variable. When taking the partial derivative of a function by a certain variable, all other variables are held constant.) A light wave consists of a very large number of photons, so the quantity<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-158-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|E(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">can interpreted as a probability density of finding a single photon at a particular point in space (for example, on a viewing screen).<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902194849\">There are many solutions to this equation. One solution of particular importance is<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902118559\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-159-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-159-Frame\"><span class=\"MathJax_MathContainer\"><span>E(x,t)=Asin(kx\u2212\u03c9t),<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.18]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the amplitude of the electric field,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">k<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the wave number, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-160-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c9<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the angular frequency. Combing this equation with<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.17<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">gives<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902186856\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-161-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-161-Frame\"><span class=\"MathJax_MathContainer\"><span>k2=\u03c92c2.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.19]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">According to de Broglie\u2019s equations, we have<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-162-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">p=\u210fk<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-163-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E=\u210f\u03c9<\/span><\/span><span style=\"font-size: 14pt\">. Substituting these equations in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.19<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">gives<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170904152351\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-164-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-164-Frame\"><span class=\"MathJax_MathContainer\"><span>p=Ec,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.20]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">or<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170904252568\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-165-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-165-Frame\"><span class=\"MathJax_MathContainer\"><span>E=pc.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.21]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Therefore, according to Einstein\u2019s general energy-momentum equation (Equation 1.11),<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.17<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">describes a particle with a zero rest mass. This is consistent with our knowledge of a photon.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170904153453\">This process can be reversed. We can begin with the energy-momentum equation of a particle and then ask what wave equation corresponds to it. The energy-momentum equation of a nonrelativistic particle in one dimension is<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903900466\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-166-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-166-Frame\"><span class=\"MathJax_MathContainer\"><span>E=p22m+U(x,t),<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.22]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">p<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the momentum,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">m<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the mass, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">U<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the potential energy of the particle. The wave equation that goes with it turns out to be a key equation in quantum mechanics, called<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term301\" style=\"font-size: 14pt\">Schr\u04e7dinger\u2019s time-dependent equation<\/span><span style=\"font-size: 14pt\">.<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1170903798652\" class=\"ui-has-child-title\">\n<header>\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\" id=\"13370\">THE SCHR\u04e6DINGER TIME-DEPENDENT EQUATION<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1170904065074\">The equation describing the energy and momentum of a wave function is known as the Schr\u04e7dinger equation:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902110094\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-167-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-167-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22m\u22022\u03a8(x,t)\u2202x2+U(x,t)\u03a8(x,t)=i\u210f\u2202\u03a8(x,t)\u2202t.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.23]<\/span><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1170902089474\">As described in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:eb80e8be-e956-45ae-b17e-ea0c66d4bbb8\" data-page=\"1\">Potential Energy and Conservation of Energy<\/a>, the force on the particle described by this equation is given by<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902179935\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-168-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-168-Frame\"><span class=\"MathJax_MathContainer\"><span>F=\u2212\u2202U(x,t)\u2202x.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.24]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">This equation plays a role in quantum mechanics similar to Newton\u2019s second law in classical mechanics. Once the potential energy of a particle is specified\u2014or, equivalently, once the force on the particle is specified\u2014we can solve this differential equation for the wave function. The solution to Newton\u2019s second law equation (also a differential equation) in one dimension is a function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">) that specifies where an object is at any time<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">. The solution to Schr\u04e7dinger\u2019s time-dependent equation provides a tool\u2014the wave function\u2014that can be used to determine where the particle is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">likely<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to be. This equation can be also written in two or three dimensions. Solving Schr\u04e7dinger\u2019s time-dependent equation often requires the aid of a computer.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902061967\">Consider the special case of a free particle. A free particle experiences no force<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-169-Frame\"><span class=\"MathJax_MathContainer\"><span>(F=0).<\/span><\/span><\/span><span>\u00a0<\/span>Based on<span>\u00a0<\/span>Equation 3.24, this requires only that<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902195526\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-170-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-170-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x,t)=U0=constant.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.25]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">For simplicity, we set<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-171-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0=0<\/span><\/span><span style=\"font-size: 14pt\">. Schr\u04e7dinger\u2019s equation then reduces to<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170904052586\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-172-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-172-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22m\u22022\u03a8(x,t)\u2202x2=i\u210f\u2202\u03a8(x,t)\u2202t.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.26]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">A valid solution to this equation is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902196880\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-173-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-173-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Aei(kx\u2212\u03c9t).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.27]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Not surprisingly, this solution contains an<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term302\" style=\"font-size: 14pt\">imaginary number<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-174-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">(i=\u22121)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">because the differential equation itself contains an imaginary number. As stressed before, however, quantum-mechanical predictions depend only on<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-175-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03a8(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">, which yields completely real values. Notice that the real plane-wave solutions,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-176-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=Asin(kx\u2212\u03c9t)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-177-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=Acos(kx\u2212\u03c9t),<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">do not obey Schr\u00f6dinger\u2019s equation. The temptation to think that a wave function can be seen, touched, and felt in nature is eliminated by the appearance of an imaginary number. In Schr\u04e7dinger\u2019s theory of quantum mechanics, the wave function is merely a tool for calculating things.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902183058\">If the potential energy function (<em data-effect=\"italics\">U<\/em>) does not depend on time, it is possible to show that<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-178-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=\u03c8(x)e\u2212i\u03c9t<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.28]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902089128\">satisfies Schr\u04e7dinger\u2019s time-dependent equation, where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-179-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>is a<span>\u00a0<\/span><em data-effect=\"italics\">time<\/em>-independent function and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-180-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u2212i\u03c9t<\/span><\/span><\/span><span>\u00a0<\/span>is a<span>\u00a0<\/span><em data-effect=\"italics\">space<\/em>-independent function. In other words, the wave function is<span>\u00a0<\/span><em data-effect=\"italics\">separable<\/em><span>\u00a0<\/span>into two parts: a space-only part and a time-only part. The factor<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-181-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u2212i\u03c9t<\/span><\/span><\/span><span>\u00a0<\/span>is sometimes referred to as a<span>\u00a0<\/span><span data-type=\"term\" id=\"term303\">time-modulation factor<\/span><span>\u00a0<\/span>since it modifies the space-only function. According to de Broglie, the energy of a matter wave is given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-182-Frame\"><span class=\"MathJax_MathContainer\"><span>E=\u210f\u03c9<\/span><\/span><\/span>, where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is its total energy. Thus, the above equation can also be written as<\/p>\n<div data-type=\"equation\" id=\"fs-id1170904230883\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-183-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-183-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=\u03c8(x)e\u2212iEt\/\u210f.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.29]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Any linear combination of such states (mixed state of energy or momentum) is also valid solution to this equation. Such states can, for example, describe a localized particle (see<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.9)<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.5<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header><span style=\"font-size: 1rem\">A particle with mass<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">m<\/em><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">is moving along the<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"font-size: 1rem\">-axis in a potential given by the potential energy function<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-184-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">U(x)=0.5m\u03c92&#215;2<\/span><\/span><span style=\"font-size: 1rem\">. Compute the product<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-185-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)*U(x)\u03a8(x,t).<\/span><\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span style=\"font-size: 1rem\">Express your answer in terms of the time-independent wave function,<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-186-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c8(x).<\/span><\/span><\/header>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Combining<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.23<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.28, Schr\u00f6dinger\u2019s time-dependent equation reduces to<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-187-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8(x)dx2+U(x)\u03c8(x)=E\u03c8(x),<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.30]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170904202033\">where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the total energy of the particle (a real number). This equation is called<span>\u00a0<\/span><span data-type=\"term\" id=\"term304\">Schr\u04e7dinger\u2019s time-independent equation<\/span>. Notice that we use \u201cbig psi\u201d<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-188-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03a8)<\/span><\/span><\/span><span>\u00a0<\/span>for the time-dependent wave function and \u201clittle psi\u201d<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-189-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03c8)<\/span><\/span><\/span><span>\u00a0<\/span>for the time-independent wave function. The wave-function solution to this equation must be multiplied by the time-modulation factor to obtain the time-dependent wave function.<\/p>\n<p id=\"fs-id1170904272107\">In the next sections, we solve Schr\u04e7dinger\u2019s time-independent equation for three cases: a quantum particle in a box, a simple harmonic oscillator, and a quantum barrier. These cases provide important lessons that can be used to solve more complicated systems. The time-independent wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-190-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>solutions must satisfy three conditions:<\/p>\n<ul id=\"fs-id1170902363308\" data-bullet-style=\"bullet\">\n<li><span class=\"MathJax_MathML\" id=\"MathJax-Element-191-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>must be a continuous function.<\/li>\n<li>The first derivative of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-192-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>with respect to space,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-193-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8(x)\/dx<\/span><\/span><\/span>, must be continuous, unless<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-194-Frame\"><span class=\"MathJax_MathContainer\"><span>V(x)=\u221e<\/span><\/span><\/span>.<\/li>\n<li><span class=\"MathJax_MathML\" id=\"MathJax-Element-195-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)<\/span><\/span><\/span><span>\u00a0<\/span>must not diverge (\u201cblow up\u201d) at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-196-Frame\"><span class=\"MathJax_MathContainer\"><span>x=\u00b1\u221e.<\/span><\/span><\/span><\/li>\n<\/ul>\n<p id=\"fs-id1170902216092\">The first condition avoids sudden jumps or gaps in the wave function. The second condition requires the wave function to be smooth at all points, except in special cases. (In a more advanced course on quantum mechanics, for example, potential spikes of infinite depth and height are used to model solids). The third condition requires the wave function be normalizable. This third condition follows from Born\u2019s interpretation of quantum mechanics. It ensures that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-197-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8(x)|2<\/span><\/span><\/span><span>\u00a0<\/span>is a finite number so we can use it to calculate probabilities.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.6<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header>\n<div class=\"os-title\"><span style=\"font-size: 1rem\">Which of the following wave functions is a valid wave-function solution for Schr\u04e7dinger\u2019s equation?<\/span><\/div>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div class=\"os-hasSolution unnumbered\" data-type=\"exercise\" id=\"fs-id1170902214402\">\n<section>\n<div data-type=\"problem\" id=\"fs-id1170903855855\">\n<div class=\"os-problem-container\">\n<p><span data-alt=\"Three graphs of Psi of x versus x are shown. The first rises then drops discontinuously to a lower value, rises again and then has a constant value. The second function looks like a breaking wave, with a crest overtaking the base. The third increases exponentially to infinity.\" data-type=\"media\" id=\"fs-id1170904111214\"><img loading=\"lazy\" decoding=\"async\" alt=\"Three graphs of Psi of x versus x are shown. The first rises then drops discontinuously to a lower value, rises again and then has a constant value. The second function looks like a breaking wave, with a crest overtaking the base. The third increases exponentially to infinity.\" data-media-type=\"image\/jpeg\" id=\"5457\" src=\"https:\/\/cnx.org\/resources\/a98eb833bef078ae0a706e5084d8e2649442cc72\" class=\"alignnone\" width=\"649\" height=\"154\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\n","protected":false},"author":615,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"3. 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