{"id":187,"date":"2019-04-09T00:25:16","date_gmt":"2019-04-09T04:25:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&#038;p=187"},"modified":"2019-04-16T12:31:59","modified_gmt":"2019-04-16T16:31:59","slug":"3-1-wave-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/3-1-wave-functions\/","title":{"raw":"3.1 Wave Functions","rendered":"3.1 Wave Functions"},"content":{"raw":"<div data-type=\"abstract\" id=\"52058\" 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 statistical interpretation of the wave function<\/li>\r\n \t<li>Use the wave function to determine probabilities<\/li>\r\n \t<li>Calculate expectation values of position, momentum, and kinetic energy<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<section id=\"fs-id1170903851901\" data-depth=\"1\">\r\n<p id=\"fs-id1170902118559\">In the preceding chapter, we saw that particles act in some cases like particles and in other cases like waves. But what does it mean for a particle to \u201cact like a wave\u201d? What precisely is \u201cwaving\u201d? What rules govern how this wave changes and propagates? How is the wave function used to make predictions? For example, if the amplitude of an electron wave is given by a function of position and time,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-1-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span>, defined for all<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">where<\/em><span>\u00a0<\/span>exactly is the electron? The purpose of this chapter is to answer these questions.<\/p>\r\n\r\n<section id=\"fs-id1170902275063\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Using the Wave Function<\/h3>\r\n<p id=\"fs-id1170902297840\">A clue to the physical meaning of the wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-2-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span><span>\u00a0<\/span>is provided by the<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term280\">two-slit interference<\/span><span>\u00a0<\/span>of monochromatic light (Figure 3.2). (See also<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:dd738e1e-40be-4540-b114-e1a24eee0ca6\" data-page=\"1\">Electromagnetic Waves<\/a><span>\u00a0<\/span>and<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:96bc01c2-ae55-4563-8976-8d66a089b8d2@5\" data-page=\"21\">Interference<\/a>.) The<span>\u00a0<\/span><span data-type=\"term\" id=\"term281\">wave function<\/span><span>\u00a0<\/span>of a light wave is given by<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em>(<em data-effect=\"italics\">x<\/em>,<em data-effect=\"italics\">t<\/em>), and its energy density is given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-3-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span>, where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the electric field strength. The energy of an individual photon depends only on the frequency of light,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-4-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b5photon=hf,<\/span><\/span><\/span><span>\u00a0<\/span>so<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-5-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span><span>\u00a0<\/span>is proportional to the number of photons. When light waves from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-6-Frame\"><span class=\"MathJax_MathContainer\"><span>S1<\/span><\/span><\/span><span>\u00a0<\/span>interfere with light waves from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-7-Frame\"><span class=\"MathJax_MathContainer\"><span>S2<\/span><\/span><\/span><span>\u00a0<\/span>at the viewing screen (a distance<span>\u00a0<\/span><em data-effect=\"italics\">D<\/em><span>\u00a0<\/span>away), an interference pattern is produced (part (a) of the figure). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (part (b)).<\/p>\r\n<p id=\"fs-id1170902109393\">Suppose the screen is initially unexposed to light. If the screen is exposed to very weak light, the interference pattern appears gradually (Figure 3.2(c), left to right). Individual photon hits on the screen appear as dots. The dot density is expected to be large at locations where the interference pattern will be, ultimately, the most intense. In other words, the probability (per unit area) that a single photon will strike a particular spot on the screen is proportional to the square of the total electric field,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-8-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span><span>\u00a0<\/span>at that point. Under the right conditions, the same interference pattern develops for matter particles, such as electrons.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_TwoSlit\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"979\"]<img alt=\"Part a shows monochromatic light of wavelength lambda emitted from a source, arriving as plane waves at a single slit, S. The waves pass through the slit ad form circular waves that arrive at a double slit, S sub 1 and S sub 2. The light rays emerge from two slits as semicircles overlapping one another. The interacting waves spread out and end on a screen where points of maximum, where the crests or troughs overlap, and minimum, where the crests from one slit overlap the troughs from the other, are marked. The pattern appears on the screen as a series of alternating bright and dark fringes. The fringes separation, y, is the distance between adjacent maxima. In part b, a photograph of the fringe pattern is shown. Part c shows how the pattern develops in time. Photos of the image at five times are shown. At first, only a few scattered bright points appear, apparently randomly, against a dark background. In the second image, we see more dots but not yet any discernible pattern. In the third image, we start to see that there are more dots in some parts of the image and fewer elsewhere. Vertical stripes of dense bright dots separated are clearly seen in the fourth image, and even more clearly in the fifth.\" data-media-type=\"image\/jpeg\" id=\"78504\" src=\"https:\/\/cnx.org\/resources\/711b4fa834522c8f4eec045173244f36080e5e48\" width=\"979\" height=\"837\" \/> Figure 3.2 Two-slit interference of monochromatic light. (a) Schematic of two-slit interference; (b) light interference pattern; (c) interference pattern built up gradually under low-intensity light (left to right).[\/caption]<\/figure>\r\n<\/div>\r\n<div class=\"media-2 ui-has-child-title\" data-type=\"note\" id=\"fs-id1170904168862\"><header>\r\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">INTERACTIVE<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1170902270676\">Visit this<span>\u00a0<\/span><a href=\"https:\/\/openstax.org\/l\/21intquawavint\" rel=\"nofollow\">interactive simulation<\/a><span>\u00a0<\/span>to learn more about quantum wave interference.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1170903831619\">The square of the matter wave<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-9-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03a8|2<\/span><\/span><\/span><span>\u00a0<\/span>in one dimension has a similar interpretation as the square of the electric field<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-10-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span>. It gives the probability that a particle will be found at a particular position and time per unit length, also called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term282\">probability density<\/span>. The probability (<em data-effect=\"italics\">P<\/em>) a particle is found in a narrow interval (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">x + dx<\/em>) at time<span>\u00a0<\/span><em data-effect=\"italics\">t<\/em><span>\u00a0<\/span>is therefore<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170904030971\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-11-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-11-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.1]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">(Later, we define the magnitude squared for the general case of a function with \u201cimaginary parts.\u201d) This probabilistic interpretation of the wave function is called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term283\" style=\"font-size: 14pt\">Born interpretation<\/span><span style=\"font-size: 14pt\">. Examples of wave functions and their squares for a particular time<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">are given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.3.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Prob_Square\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"486\"]<img alt=\"Three wave functions and the square of the amplitude of the wave functions. The first wave function, psi sub zero, and its square are symmetric, positive, peaked at x = 0, and zero far from the origin. The second wave function, psi sub 1, is antisymmetric: the function is negative at negative x, positive at positive x, and zero at the origin as well as plus and minus infinity. There is a negative minimum at negative x and positive maximum at positive x. The minimum value is exactly opposite the maximum value. The square of the amplitude of the wave function is positive and symmetric about the origin, where the value is zero, with a maximum on either side of the origin. The third wave function, psi sub N, is not symmetric. It is zero at minus infinity, decreases to a negative minimum value at some x less than zero, crosses zero, still at x less than zero, and becomes positive. It reaches a positive maximum at some positive x, then decreases to zero at large x. The minimum value is smaller in magnitude than the maximum value. The square of the amplitude of the wave function is positive, with two local maxima. The first maximum is smaller and at negative x, and the second is larger and at positive x. The square of the function is zero at one point between the maxima.\" data-media-type=\"image\/jpeg\" id=\"56881\" src=\"https:\/\/cnx.org\/resources\/2d838f6061b1af999544170bcc02c79688d82134\" width=\"486\" height=\"469\" \/> Figure 3.3 Several examples of wave functions and the corresponding square of their wave functions.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170903805831\">If the wave function varies slowly over the interval<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-12-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0394x<\/span><\/span><\/span>, the probability a particle is found in the interval is approximately<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902089475\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-13-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-13-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248|\u03a8(x,t)|2\u0394x.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.2]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Notice that squaring the wave function ensures that the probability is positive. (This is analogous to squaring the electric field strength\u2014which may be positive or negative\u2014to obtain a positive value of intensity.) However, if the wave function does not vary slowly, we must integrate:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170903900559\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-14-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-14-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)=\u222bxx+\u0394x|\u03a8(x,t)|2dx.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.3]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">This probability is just the area under the function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-15-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03a8(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><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-16-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">x+\u0394x<\/span><\/span><span style=\"font-size: 14pt\">. The probability of finding the particle \u201csomewhere\u201d (the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term284\" style=\"font-size: 14pt\">normalization condition<\/span><span style=\"font-size: 14pt\">) is<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-17-Frame\"><span class=\"MathJax_MathContainer\"><span>P(\u2212\u221e,+\u221e)=\u222b\u2212\u221e\u221e|\u03a8(x,t)|2dx=1.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.4]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">For a particle in two dimensions, the integration is over an area and requires a double integral; for a particle in three dimensions, the integration is over a volume and requires a triple integral. For now, we stick to the simple one-dimensional case.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1170902178031\" class=\"ui-has-child-title\"><header><\/header><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\"><\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170902089222\"><span data-type=\"title\"><strong>Where Is the Ball? (Part I)<\/strong><\/span><\/p>\r\nA ball is constrained to move along a line inside a tube of length<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>. The ball is equally likely to be found anywhere in the tube at some time<span>\u00a0<\/span><em data-effect=\"italics\">t<\/em>. What is the probability of finding the ball in the left half of the tube at that time? (The answer is 50%, of course, but how do we get this answer by using the probabilistic interpretation of the quantum mechanical wave function?)\r\n\r\n<span data-type=\"title\"><strong>Strategy<\/strong><\/span>\r\n\r\nThe first step is to write down the wave function. The ball is equally like to be found anywhere in the box, so one way to describe the ball with a<span>\u00a0<\/span><em data-effect=\"italics\">constant<\/em><span>\u00a0<\/span>wave function (Figure 3.4). The normalization condition can be used to find the value of the function and a simple integration over half of the box yields the final answer.\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Square_Wave\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"465\"]<img alt=\"The wave function Psi of x and t is plotted as a function of x. It is a step function, zero for x less than 0 and x greater than L, and constant for x between zero and L.\" data-media-type=\"image\/jpeg\" id=\"22284\" src=\"https:\/\/cnx.org\/resources\/1db9b1f532811d28b55a78a01f666df7b0461fef\" width=\"465\" height=\"284\" \/> Figure 3.4 Wave function for a ball in a tube of length L.[\/caption]<\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\">\r\n<strong>Solution<\/strong><\/span><\/span><\/span><\/span>\r\n\r\n<span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">The wave function of the ball can be written as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-18-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=C(0&lt;x&lt;L),<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is a constant, and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-19-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">otherwise. We can determine the constant<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">by applying the normalization condition (we set<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-20-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">t=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to simplify the notation):<\/span>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902141124\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-21-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x=\u2212\u221e,+\u221e)=\u222b\u2212\u221e\u221e|C|2dx=1.\r\n<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\r\nThis integral can be broken into three parts: (1) negative infinity to zero, (2) zero to<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">, and (3)<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to infinity. The particle is constrained to be in the tube, so<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-22-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">C=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">outside the tube and the first and last integrations are zero. The above equation can therefore be written<\/span><span>\r\n<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904208604\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-23-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>P(x=0,L)=\u222b0L|C|2dx=1.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The value<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">does not depend on<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and can be taken out of the integral, so we obtain<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903857790\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-24-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>|C|2\u222b0Ldx=1.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Integration gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902289643\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-25-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>C=1L.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">To determine the probability of finding the ball in the first half of the box<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-26-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(0&lt;x&lt;L),<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">we have<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904134011\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-27-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>P(x=0,L\/2)=\u222b0L\/2|1L|2dx=(1L)L2=0.50.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The probability of finding the ball in the first half of the tube is 50%, as expected. Two observations are noteworthy. First, this result corresponds to the area under the constant function from<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-28-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2 (the area of a square left of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2). Second, this calculation requires an integration of the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">square<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">of the wave function. A common mistake in performing such calculations is to forget to square the wave function before integration.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div data-type=\"example\" id=\"fs-id1170904209524\" class=\"ui-has-child-title\"><section>\r\n<div class=\"textbox shaded\">\r\n<div data-type=\"example\" id=\"fs-id1170902178031\" class=\"ui-has-child-title\"><section><span class=\"os-title-label\" style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">EXAMPLE 3<\/span><span class=\"os-number\" style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">.2<\/span><\/section><\/div>\r\n<div data-type=\"example\" id=\"fs-id1170904209524\" class=\"ui-has-child-title\"><header>\r\n<p class=\"os-title\"><strong>Where Is the Ball? (Part II)<\/strong><\/p>\r\n<span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">A ball is again constrained to move along a line inside a tube of length<\/span><span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">\u00a0<\/span><em style=\"font-family: Tinos, Georgia, serif;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">. This time, the ball is found preferentially in the middle of the tube. One way to represent its wave function is with a simple cosine function (<\/span>Figure 3.5<span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">). What is the probability of finding the ball in the last one-quarter of the tube?<\/span>\r\n<span class=\"os-divider\"><\/span>\r\n\r\n<\/header><section>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Cosine_Wave\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"426\"]<img alt=\"A plot of Psi of x at t equal to zero as a function of x. The function is zero for x less than minus L over 2 and x greater than L over 2. For x between minus and plus L over 2, the functions is a cosine curve, concave down with a positive maximum at x equal to zero, and going to zero at minus and plus L over 2.\" data-media-type=\"image\/jpeg\" id=\"93531\" src=\"https:\/\/cnx.org\/resources\/07ab66f742ac10c60f8b3ae4bb8c4efcfb52521a\" width=\"426\" height=\"295\" \/> Figure 3.5 Wave function for a ball in a tube of length L, where the ball is preferentially in the middle of the tube.<span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><\/span><\/span><\/span><br \/><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><\/span><\/span><\/span>[\/caption]<\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\">\r\n<strong>Strategy<\/strong><\/span><\/span><\/span><\/span>\r\n\r\n<span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">We use the same strategy as before. In this case, the wave function has two unknown constants: One is associated with the wavelength of the wave and the other is the amplitude of the wave. We determine the amplitude by using the boundary conditions of the problem, and we evaluate the wavelength by using the normalization condition. Integration of the square of the wave function over the last quarter of the tube yields the final answer. The calculation is simplified by centering our coordinate system on the peak of the wave function.\r\n\r\n<\/span><\/span><\/span><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span><\/span><\/span>\r\n\r\n<span style=\"font-size: 1rem;text-indent: 1em\">The wave function of the ball can be written<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903814492\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-29-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03a8(x,0)=Acos(kx)(\u2212L\/2&lt;x&lt;L\/2),<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">A<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the amplitude of the wave function and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-30-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">k=2\u03c0\/\u03bb<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is its wave number. Beyond this interval, the amplitude of the wave function is zero because the ball is confined to the tube. Requiring the wave function to terminate at the right end of the tube gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902216824\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-31-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03a8(x=L2,0)=0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Evaluating the wave function at<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-32-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L\/2<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">gives<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903906927\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-33-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>Acos(kL\/2)=0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">This equation is satisfied if the argument of the cosine is an integral multiple of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-34-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c0\/2,3\u03c0\/2,5\u03c0\/2,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and so on. In this case, we have<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902132492\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-35-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>kL2=\u03c02,<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">or<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903756446\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-36-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>k=\u03c0L.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Applying the normalization condition gives<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-37-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">A=2\/L<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so the wave function of the ball is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904271710\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-38-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03a8(x,0)=2Lcos(\u03c0x\/L),\u2212L\/2&lt;x&lt;L\/2.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">To determine the probability of finding the ball in the last quarter of the tube, we square the function and integrate:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902339022\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-39-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>P(x=L\/4,L\/2)=\u222bL\/4L\/2|2Lcos(\u03c0xL)|2dx=0.091.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The probability of finding the ball in the last quarter of the tube is 9.1%. The ball has a definite wavelength<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-40-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(\u03bb=2L)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. If the tube is of macroscopic length<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-41-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(L=1m)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, the momentum of the ball is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902275088\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-42-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>p=h\u03bb=h2L~10\u221236m\/s.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">This momentum is much too small to be measured by any human instrument.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<span style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">An Interpretation of the Wave Function<\/span>\r\n\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1170902128957\" data-depth=\"1\">\r\n<p id=\"fs-id1170904168306\">We are now in position to begin to answer the questions posed at the beginning of this section. First, for a traveling particle described by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-43-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Asin(kx\u2212\u03c9t)<\/span><\/span><\/span>, what is \u201cwaving?\u201d Based on the above discussion, the answer is a mathematical function that can, among other things, be used to determine where the particle is likely to be when a position measurement is performed. Second, how is the wave function used to make predictions? If it is necessary to find the probability that a particle will be found in a certain interval, square the wave function and integrate over the interval of interest. Soon, you will learn soon that the wave function can be used to make many other kinds of predictions, as well.<\/p>\r\n<p id=\"fs-id1170904071853\">Third, if a matter wave is given by the wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-44-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">where<\/em><span>\u00a0<\/span>exactly is the particle? Two answers exist: (1) when the observer<span>\u00a0<\/span><em data-effect=\"italics\">is not<\/em><span>\u00a0<\/span>looking (or the particle is not being otherwise detected), the particle is everywhere<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-45-Frame\"><span class=\"MathJax_MathContainer\"><span>(x=\u2212\u221e,+\u221e)<\/span><\/span><\/span>; and (2) when the observer<span>\u00a0<\/span><em data-effect=\"italics\">is<\/em><span>\u00a0<\/span>looking (the particle is being detected), the particle \u201cjumps into\u201d a particular position state<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-46-Frame\"><span class=\"MathJax_MathContainer\"><span>(x,x+dx)<\/span><\/span><\/span><span>\u00a0<\/span>with a probability given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-47-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx<\/span><\/span><\/span><span>\u00a0<\/span>\u2014a process called<span>\u00a0<\/span><span data-type=\"term\" id=\"term285\">state reduction<\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span data-type=\"term\" id=\"term286\">wave function collapse<\/span>. This answer is called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term287\">Copenhagen interpretation<\/span><span>\u00a0<\/span>of the wave function, or of quantum mechanics.<\/p>\r\n<p id=\"fs-id1170902247796\">To illustrate this interpretation, consider the simple case of a particle that can occupy a small container either at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-48-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-49-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>(Figure 3.6). In classical physics, we assume the particle is located either at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-50-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-51-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>when the observer is not looking. However, in quantum mechanics, the particle may exist in a state of indefinite position\u2014that is, it may be located at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-52-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span><em data-effect=\"italics\">and<\/em><span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-53-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>when the observer is not looking. The assumption that a particle can only have one value of position (when the observer is not looking) is abandoned. Similar comments can be made of other measurable quantities, such as momentum and energy.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Two_State\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"500\"]<img alt=\"An x y coordinate system is shown with two small boxes drawn on the x axis, one at x sub 1 to the left of the origin and the other at x sub 2 to the right of the origin.\" data-media-type=\"image\/jpeg\" id=\"73184\" src=\"https:\/\/cnx.org\/resources\/139a50ec55ffda7dd8f33fe25a5b172e28353f54\" width=\"500\" height=\"172\" \/> Figure 3.6 A two-state system of position of a particle.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170903864386\">The bizarre consequences of the Copenhagen interpretation of quantum mechanics are illustrated by a creative thought experiment first articulated by Erwin Schr\u00f6dinger (<em data-effect=\"italics\">National Geographic<\/em>, 2013) (Figure 3.7):<\/p>\r\n<p id=\"fs-id1170903875163\">\u201cA cat is placed in a steel box along with a Geiger counter, a vial of poison, a hammer, and a radioactive substance. When the radioactive substance decays, the Geiger detects it and triggers the hammer to release the poison, which subsequently kills the cat. The radioactive decay is a random [probabilistic] process, and there is no way to predict when it will happen. Physicists say the atom exists in a state known as a superposition\u2014both decayed and not decayed at the same time. Until the box is opened, an observer doesn\u2019t know whether the cat is alive or dead\u2014because the cat\u2019s fate is intrinsically tied to whether or not the atom has decayed and the cat would [according to the Copenhagen interpretation] be \u201cliving and dead ... in equal parts\u201d until it is observed.\u201d<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Cat\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"974\"]<img alt=\"The Schodinger cat thought experiment, consisting of a Geiger counter, a vial of poison, a hammer, a radioactive substance, and a cat is illustrated. Each is shown in two states: The Geiger counter triggered and untriggered, the hammer up and down, the poison vial whole and broken, and the cat alive and dead.\" data-media-type=\"image\/jpeg\" id=\"93175\" src=\"https:\/\/cnx.org\/resources\/ba43497f5351605d6b1cf05c2744d5852ede67ca\" width=\"974\" height=\"518\" \/> Figure 3.7 Schr\u00f6dinger\u2019s cat. Schr\u00f6dinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170903874503\">Two-state systems (left and right, atom decays and does not decay, and so on) are often used to illustrate the principles of quantum mechanics. These systems find many applications in nature, including electron spin and mixed states of particles, atoms, and even molecules. Two-state systems are also finding application in the quantum computer, as mentioned in the introduction of this chapter. Unlike a digital computer, which encodes information in binary digits (zeroes and ones), a quantum computer stores and manipulates data in the form of quantum bits, or qubits. In general, a<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term289\">qubit<\/span><span>\u00a0<\/span>is not in a state of zero or one, but rather in a mixed state of zero<span>\u00a0<\/span><em data-effect=\"italics\">and<\/em>one. If a large number of qubits are placed in the same quantum state, the measurement of an individual qubit would produce a zero with a probability<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>, and a one with a probability<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-54-Frame\"><span class=\"MathJax_MathContainer\"><span>q=1\u2212p.<\/span><\/span><\/span><span>\u00a0<\/span>Many scientists believe that quantum computers are the future of the computer industry.<\/p>\r\n\r\n<\/section><section id=\"fs-id1170902337169\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Complex Conjugates<\/h3>\r\n<p id=\"fs-id1170903874497\">Later in this section, you will see how to use the wave function to describe particles that are \u201cfree\u201d or bound by forces to other particles. The specific form of the wave function depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually\u00a0<span data-type=\"term\" id=\"term290\">complex function<\/span><strong data-effect=\"bold\">s<\/strong>. A complex function is one that contains one or more imaginary numbers\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-55-Frame\"><span class=\"MathJax_MathContainer\"><span>(i=\u22121)<\/span><\/span><\/span>. Experimental measurements produce real (nonimaginary) numbers only, so the above procedure to use the wave function must be slightly modified. In general, the probability that a particle is found in the narrow interval (<em data-effect=\"italics\">x<\/em>,<em data-effect=\"italics\">x + dx<\/em>) at time<em data-effect=\"italics\">t<\/em>is given by<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-56-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx=\u03a8*(x,t)\u03a8(x,t)dx,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.5]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902243906\">where<span class=\"MathJax_MathML\" id=\"MathJax-Element-57-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)<\/span><\/span><\/span>is the complex conjugate of the wave function. The complex conjugate of a function is obtaining by replacing every occurrence of<span class=\"MathJax_MathML\" id=\"MathJax-Element-58-Frame\"><span class=\"MathJax_MathContainer\"><span>i=\u22121<\/span><\/span><\/span>in that function with<span class=\"MathJax_MathML\" id=\"MathJax-Element-59-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212i<\/span><\/span><\/span>. This procedure eliminates complex numbers in all predictions because the product<span class=\"MathJax_MathML\" id=\"MathJax-Element-60-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)\u03a8(x,t)<\/span><\/span><\/span>is always a real number.<\/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\">.1<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header><span style=\"font-size: 1rem\">If<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-61-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">a=3+4i<\/span><\/span><span style=\"font-size: 1rem\">, what is the product<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-62-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">a*a<\/span><\/span><span style=\"font-size: 1rem\">?<\/span><\/header><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903874962\">Consider the motion of a free particle that moves along the\u00a0<em data-effect=\"italics\">x<\/em>-direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wave function has real\u00a0<em data-effect=\"italics\">and\u00a0<\/em>complex parts. In particular, the wave function is given by<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903886955\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-63-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Acos(kx\u2212\u03c9t)+iAsin(kx\u2212\u03c9t),<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170904060990\">where\u00a0<em data-effect=\"italics\">A\u00a0<\/em>is the amplitude,\u00a0<em data-effect=\"italics\">k\u00a0<\/em>is the wave number, and<span class=\"MathJax_MathML\" id=\"MathJax-Element-64-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c9<\/span><\/span><\/span>is the angular frequency. Using Euler\u2019s formula,<span class=\"MathJax_MathML\" id=\"MathJax-Element-65-Frame\"><span class=\"MathJax_MathContainer\"><span>ei\u03d5=cos(\u03d5)+isin(\u03d5),<\/span><\/span><\/span>this equation can be written in the form<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902187505\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-66-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Aei(kx\u2212\u03c9t)=Aei\u03d5,<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902097748\">where<span class=\"MathJax_MathML\" id=\"MathJax-Element-67-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span>is the phase angle. If the wave function varies slowly over the interval<span class=\"MathJax_MathML\" id=\"MathJax-Element-68-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0394x,<\/span><\/span><\/span>the probability of finding the particle in that interval is<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902214038\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-69-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248\u03a8*(x,t)\u03a8(x,t)\u0394x=(Aei\u03d5)(A*e\u2212i\u03d5)\u0394x=(A*A)\u0394x.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902041507\">If<em data-effect=\"italics\">A<\/em>has real and complex parts<span class=\"MathJax_MathML\" id=\"MathJax-Element-70-Frame\"><span class=\"MathJax_MathContainer\"><span>(a+ib<\/span><\/span><\/span>, where\u00a0<em data-effect=\"italics\">a\u00a0<\/em>and\u00a0<em data-effect=\"italics\">b\u00a0<\/em>are real constants), then<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903751129\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-71-Frame\"><span class=\"MathJax_MathContainer\"><span>A*A=(a+ib)(a\u2212ib)=a2+b2.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902091680\">Notice that the complex numbers have vanished. Thus,<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902300451\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-72-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248|A|2\u0394x<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903832048\">is a real quantity. The interpretation of<span class=\"MathJax_MathML\" id=\"MathJax-Element-73-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)\u03a8(x,t)<\/span><\/span><\/span>as a probability density ensures that the predictions of quantum mechanics can be checked in the \u201creal world.\u201d<\/p>\r\n\r\n<\/section>\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\">.2<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header><span style=\"font-size: 1rem\">Suppose that a particle with energy<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">E<\/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 and is confined in the region between 0 and<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">. One possible wave function is<\/span><\/header><section>\r\n<div class=\"os-note-body\">\r\n<div class=\"os-hasSolution unnumbered\" data-type=\"exercise\" id=\"fs-id1170904134843\"><section>\r\n<div data-type=\"problem\" id=\"fs-id1170902114584\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1172100931629\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-74-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x,t)={Ae\u2212iEt\/\u210fsin\u03c0xL,when 0\u2264x\u2264L0,otherwise.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902189634\">Determine the normalization constant.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<span style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">Expectation Values<\/span>\r\n\r\n<section id=\"fs-id1170903851901\" data-depth=\"1\">\r\n<p id=\"fs-id1170903874481\">In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as <em data-effect=\"italics\">x<\/em>(<em data-effect=\"italics\">t<\/em>), where\u00a0<em data-effect=\"italics\">x\u00a0<\/em>is the position and\u00a0<em data-effect=\"italics\">t\u00a0<\/em>is the time. Note that the particle has one value of position for any time <em data-effect=\"italics\">t<\/em>. In quantum mechanics, however, the solution to an equation of motion is a wave function,<span class=\"MathJax_MathML\" id=\"MathJax-Element-75-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t). <\/span><\/span><\/span>The particle has many values of position for any time <em data-effect=\"italics\">t<\/em>, and only the probability density of finding the particle,<span class=\"MathJax_MathML\" id=\"MathJax-Element-76-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03a8(x,t)|2<\/span><\/span><\/span>, can be known. The average value of position for a large number of particles with the same wave function is expected to be<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170904189252\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-77-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-77-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e\u221exP(x,t)dx=\u222b\u2212\u221e\u221ex\u03a8*(x,t)\u03a8(x,t)dx.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.6]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">This is called the\u00a0<\/span><span data-type=\"term\" id=\"term291\" style=\"font-size: 14pt\">expectation value o<\/span><span style=\"font-size: 14pt\">f the position. It is usually written<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-78-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e\u221e\u03a8*(x,t)x\u03a8(x,t)dx,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.7]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where the <\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x <\/em><span style=\"font-size: 14pt\">is sandwiched between the wave functions. The reason for this will become apparent soon. Formally,<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">is called the\u00a0<\/span><span data-type=\"term\" id=\"term292\" style=\"font-size: 14pt\">position operator<\/span><span style=\"font-size: 14pt\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902235000\">At this point, it is important to stress that a wave function can be written in terms of other quantities as well, such as velocity (<em data-effect=\"italics\">v<\/em>), momentum (<em data-effect=\"italics\">p<\/em>), and kinetic energy (<em data-effect=\"italics\">K<\/em>). The expectation value of momentum, for example, can be written<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902149970\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-79-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-79-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b\u2212\u221e\u221e\u03a8*(p,t)p\u03a8(p,t)dp,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.8]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Where\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dp\u00a0<\/em><span style=\"font-size: 14pt\">is used instead of\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dx\u00a0<\/em><span style=\"font-size: 14pt\">to indicate an infinitesimal interval in momentum. In some cases, we know the wave function in position,<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-80-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t),<\/span><\/span><span style=\"font-size: 14pt\">but seek the expectation of momentum. The procedure for doing this is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902208206\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-81-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-81-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b\u2212\u221e\u221e\u03a8*(x,t)(\u2212i\u210fddx)\u03a8(x,t)dx,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.9]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where the quantity in parentheses, sandwiched between the wave functions, is called the\u00a0<\/span><span data-type=\"term\" id=\"term293\" style=\"font-size: 14pt\">momentum operator\u00a0<\/span><span style=\"font-size: 14pt\">in the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-direction. [The momentum operator in Equation 3.9is said to be the position-space representation of the momentum operator.] The momentum operator must act (operate) on the wave function to the right, and then the result must be multiplied by the complex conjugate of the wave function on the left, before integration. The momentum operator in the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-direction is sometimes denoted<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170904245770\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-82-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-82-Frame\"><span class=\"MathJax_MathContainer\"><span>(px)op=\u2212i\u210fddx,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.10]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Momentum operators for the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">- and <\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-directions are defined similarly. This operator and many others are derived in a more advanced course in modern physics. In some cases, this derivation is relatively simple. For example, the kinetic energy operator is just<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902293433\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-83-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-83-Frame\"><span class=\"MathJax_MathContainer\"><span>(K)op=12m(vx)op2=(px)op22m=(\u2212i\u210fddx)22m=\u2212\u210f22m(ddx)(ddx).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.11]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Thus, if we seek an expectation value of kinetic energy of a particle in one dimension, two successive ordinary derivatives of the wave function are required before integration.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902271392\">Expectation-value calculations are often simplified by exploiting the symmetry of wave functions. Symmetric wave functions can be even or odd. An\u00a0<span data-type=\"term\" id=\"term294\">even function\u00a0<\/span>is a function that satisfies<\/p>\r\n\r\n<div data-type=\"equation\" id=\"eip-240\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-84-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-84-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)=\u03c8(\u2212x).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.12]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In contrast, an\u00a0<\/span><span data-type=\"term\" id=\"term295\" style=\"font-size: 14pt\">odd function\u00a0<\/span><span style=\"font-size: 14pt\">is a function that satisfies<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"eip-912\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-85-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-85-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)=-\u03c8(\u2212x).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.13]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">An example of even and odd functions is shown in Figure 3.8. An even function is symmetric about the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis. This function is produced by reflecting<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-86-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">for<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">&gt; 0 about the vertical\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis. By comparison, an odd function is generated by reflecting the function about the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis and then about the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-axis. (An odd function is also referred to as an\u00a0<\/span><span data-type=\"term\" id=\"term296\" style=\"font-size: 14pt\">anti-symmetric function<\/span><span style=\"font-size: 14pt\">.)<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_01_Even_Odd\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"381\"]<img alt=\"Two wave functions are plotted as a function of x. The vertical scale runs form -0.5 to +0.5 and the horizontal scale from -4 to 4. The even function is plotted in blue. It is symmetric about the origin, positive for all values of x, and going to zero at the ends. This particular even function has a positive minimum at the origin and maxima on either side. The odd function is zero at the origin and at the ends, negative to the left of the origin, where it has a minimum, and positive to the right, where it has a maximum. The function is antisymmetric, meaning that the negative half is the same shape as the right half, but inverted, that is, generated by reflecting the function about the y-axis and then about the x-axis.\" data-media-type=\"image\/jpeg\" id=\"99542\" src=\"https:\/\/cnx.org\/resources\/e6dca3667a707a47cd782cc5794d7bbfce297779\" width=\"381\" height=\"233\" \/> Figure 3.8 Examples of even and odd wave functions.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170902236504\">In general, an even function times an even function produces an even function. A simple example of an even function is the product\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-87-Frame\"><span class=\"MathJax_MathContainer\"><span>x2e\u2212x2\u00a0<\/span><\/span><\/span>(even times even is even). Similarly, an odd function times an odd function produces an even function, such as\u00a0<em data-effect=\"italics\">x<\/em>s in\u00a0<em data-effect=\"italics\">x\u00a0<\/em>(odd times odd is even). However, an odd function times an even function produces an odd function, such as\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-88-Frame\"><span class=\"MathJax_MathContainer\"><span>xe\u2212x2<\/span><\/span><\/span>(odd times even is odd). The integral over all space of an odd function is zero, because the total area of the function above the\u00a0<em data-effect=\"italics\">x<\/em>-axis cancels the (negative) area below it. As the next example shows, this property of odd functions is very useful.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1170904072147\" class=\"ui-has-child-title\"><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.3<\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170902178244\"><span data-type=\"title\"><strong>Expectation Value (Part I)<\/strong><\/span><\/p>\r\nThe normalized wave function of a particle is\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902178250\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-89-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03c8(x)=e\u2212|x|\/x0\/x0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Find the expectation value of position.<\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Substitute the wave function into\u00a0<\/span>Equation 3.7<span style=\"text-indent: 1em;font-size: 1rem\">and evaluate. The position operator introduces a multiplicative factor only, so the position operator need not be \u201csandwiched.\u201d<\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">First multiply, then integrate:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170900367549\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-90-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e+\u221edxx|\u03c8(x)|2=\u222b\u2212\u221e+\u221edxx|e\u2212|x|\/x0x0|2=1x0\u222b\u2212\u221e+\u221edxxe\u22122|x|\/x0=0.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The function in the integrand<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-91-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\"> (xe\u22122|x|\/x0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is odd since it is the product of an odd function (<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">) and an even function<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-92-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(e\u22122|x|\/x0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. The integral vanishes because the total area of the function about the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis cancels the (negative) area below it. The result<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-93-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(\u2329x\u232a=0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is not surprising since the probability density function is symmetric about\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-94-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.4<\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170902346418\"><span data-type=\"title\"><strong>Expectation Value (Part II)<\/strong><\/span><\/p>\r\nThe time-dependent wave function of a particle confined to a region between 0 and <em data-effect=\"italics\">L <\/em>is\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902271963\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-95-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03c8(x,t)=Ae\u2212i\u03c9tsin(\u03c0x\/L)<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-96-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c9<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is angular frequency and <\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">E <\/em><span style=\"text-indent: 1em;font-size: 1rem\">is the energy of the particle. (<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">Note:<\/em><span style=\"text-indent: 1em;font-size: 1rem\">The function varies as a sine because of the limits (0 to<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">). When\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-97-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">the sine factor is zero and the wave function is zero, consistent with the boundary conditions.) Calculate the expectation values of position, momentum, and kinetic energy.<\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">We must first normalize the wave function to find\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">A<\/em><span style=\"text-indent: 1em;font-size: 1rem\">. Then we use the operators to calculate the expectation values.<\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Computation of the normalization constant:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902272294\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-98-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>1=\u222b0Ldx\u03c8*(x)\u03c8(x)=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(Ae\u2212i\u03c9tsin\u03c0xL)=A2\u222b0Ldxsin2\u03c0xL=A2L2\u21d2A=2L.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of position is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902203736\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-99-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b0Ldx\u03c8*(x)x\u03c8(x)=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)x(Ae\u2212i\u03c9tsin\u03c0xL)=A2\u222b0Ldxxsin2\u03c0xL=A2L24=L2.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of momentum in the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction also requires an integral. To set this integral up, the associated operator must\u2014 by rule\u2014act to the right on the wave function<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-100-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902344048\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-101-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2212i\u210fddx\u03c8(x)=\u2212i\u210fddxAe\u2212i\u03c9tsin\u03c0xL=\u2212iAh2Le\u2212i\u03c9tcos\u03c0xL.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Therefore, the expectation value of momentum is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902266047\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-102-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(\u2212iAh2Le\u2212i\u03c9tcos\u03c0xL)=\u2212iA2h4L\u222b0Ldxsin2\u03c0xL=0.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The function in the integral is a sine function with a wavelength equal to the width of the well,<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u2014an odd function about\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-103-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L\/2<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. As a result, the integral vanishes.<\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of kinetic energy in the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction requires the associated operator to act on the wave function:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902110544\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-104-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2dx2\u03c8(x)=\u2212\u210f22md2dx2Ae\u2212i\u03c9tsin\u03c0xL=\u2212\u210f22mAe\u2212i\u03c9td2dx2sin\u03c0xL=Ah22mL2e\u2212i\u03c9tsin\u03c0xL.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">Thus, the expectation value of the kinetic energy is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902062116\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-105-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u2329K\u232a=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(Ah22mL2e\u2212i\u03c9tsin\u03c0xL)=A2h22mL2\u222b0Ldxsin2\u03c0xL=A2h22mL2L2=h22mL2.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">The average position of a large number of particles in this state is<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2. The average momentum of these particles is zero because a given particle is equally likely to be moving right or left. However, the particle is not at rest because its average kinetic energy is not zero. Finally, the probability density is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902255479\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-106-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>|\u03c8|2=(2\/L)sin2(\u03c0x\/L).<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">This probability density is largest at location\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2 and is zero at<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-107-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">and at<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-108-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">Note that these conclusions do not depend explicitly on time.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/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\">.3<\/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\">For the particle in the above example, find the probability of locating it between positions 0 and\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">\/4<\/span><\/div>\r\n<\/header><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170904139295\">Quantum mechanics makes many surprising predictions. However, in 1920, Niels Bohr (founder of the Niels Bohr Institute in Copenhagen, from which we get the term \u201cCopenhagen interpretation\u201d) asserted that the predictions of quantum mechanics and classical mechanics must agree for all macroscopic systems, such as orbiting planets, bouncing balls, rocking chairs, and springs. This\u00a0<span data-type=\"term\" id=\"term297\">correspondence principle\u00a0<\/span>is now generally accepted. It suggests the rules of classical mechanics are an approximation of the rules of quantum mechanics for systems with very large energies. Quantum mechanics describes both the microscopic and macroscopic world, but classical mechanics describes only the latter.<\/p>\r\n\r\n<\/section>&nbsp;\r\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\r\n<\/section>","rendered":"<div data-type=\"abstract\" id=\"52058\" 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 statistical interpretation of the wave function<\/li>\n<li>Use the wave function to determine probabilities<\/li>\n<li>Calculate expectation values of position, momentum, and kinetic energy<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section id=\"fs-id1170903851901\" data-depth=\"1\">\n<p id=\"fs-id1170902118559\">In the preceding chapter, we saw that particles act in some cases like particles and in other cases like waves. But what does it mean for a particle to \u201cact like a wave\u201d? What precisely is \u201cwaving\u201d? What rules govern how this wave changes and propagates? How is the wave function used to make predictions? For example, if the amplitude of an electron wave is given by a function of position and time,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-1-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span>, defined for all<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">where<\/em><span>\u00a0<\/span>exactly is the electron? The purpose of this chapter is to answer these questions.<\/p>\n<section id=\"fs-id1170902275063\" data-depth=\"1\">\n<h3 data-type=\"title\">Using the Wave Function<\/h3>\n<p id=\"fs-id1170902297840\">A clue to the physical meaning of the wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-2-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span><span>\u00a0<\/span>is provided by the<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term280\">two-slit interference<\/span><span>\u00a0<\/span>of monochromatic light (Figure 3.2). (See also<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:dd738e1e-40be-4540-b114-e1a24eee0ca6\" data-page=\"1\">Electromagnetic Waves<\/a><span>\u00a0<\/span>and<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:96bc01c2-ae55-4563-8976-8d66a089b8d2@5\" data-page=\"21\">Interference<\/a>.) The<span>\u00a0<\/span><span data-type=\"term\" id=\"term281\">wave function<\/span><span>\u00a0<\/span>of a light wave is given by<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em>(<em data-effect=\"italics\">x<\/em>,<em data-effect=\"italics\">t<\/em>), and its energy density is given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-3-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span>, where<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the electric field strength. The energy of an individual photon depends only on the frequency of light,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-4-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b5photon=hf,<\/span><\/span><\/span><span>\u00a0<\/span>so<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-5-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span><span>\u00a0<\/span>is proportional to the number of photons. When light waves from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-6-Frame\"><span class=\"MathJax_MathContainer\"><span>S1<\/span><\/span><\/span><span>\u00a0<\/span>interfere with light waves from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-7-Frame\"><span class=\"MathJax_MathContainer\"><span>S2<\/span><\/span><\/span><span>\u00a0<\/span>at the viewing screen (a distance<span>\u00a0<\/span><em data-effect=\"italics\">D<\/em><span>\u00a0<\/span>away), an interference pattern is produced (part (a) of the figure). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (part (b)).<\/p>\n<p id=\"fs-id1170902109393\">Suppose the screen is initially unexposed to light. If the screen is exposed to very weak light, the interference pattern appears gradually (Figure 3.2(c), left to right). Individual photon hits on the screen appear as dots. The dot density is expected to be large at locations where the interference pattern will be, ultimately, the most intense. In other words, the probability (per unit area) that a single photon will strike a particular spot on the screen is proportional to the square of the total electric field,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-8-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span><span>\u00a0<\/span>at that point. Under the right conditions, the same interference pattern develops for matter particles, such as electrons.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_TwoSlit\">\n<figure style=\"width: 979px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Part a shows monochromatic light of wavelength lambda emitted from a source, arriving as plane waves at a single slit, S. The waves pass through the slit ad form circular waves that arrive at a double slit, S sub 1 and S sub 2. The light rays emerge from two slits as semicircles overlapping one another. The interacting waves spread out and end on a screen where points of maximum, where the crests or troughs overlap, and minimum, where the crests from one slit overlap the troughs from the other, are marked. The pattern appears on the screen as a series of alternating bright and dark fringes. The fringes separation, y, is the distance between adjacent maxima. In part b, a photograph of the fringe pattern is shown. Part c shows how the pattern develops in time. Photos of the image at five times are shown. At first, only a few scattered bright points appear, apparently randomly, against a dark background. In the second image, we see more dots but not yet any discernible pattern. In the third image, we start to see that there are more dots in some parts of the image and fewer elsewhere. Vertical stripes of dense bright dots separated are clearly seen in the fourth image, and even more clearly in the fifth.\" data-media-type=\"image\/jpeg\" id=\"78504\" src=\"https:\/\/cnx.org\/resources\/711b4fa834522c8f4eec045173244f36080e5e48\" width=\"979\" height=\"837\" \/><figcaption class=\"wp-caption-text\">Figure 3.2 Two-slit interference of monochromatic light. (a) Schematic of two-slit interference; (b) light interference pattern; (c) interference pattern built up gradually under low-intensity light (left to right).<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<div class=\"media-2 ui-has-child-title\" data-type=\"note\" id=\"fs-id1170904168862\">\n<header>\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">INTERACTIVE<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1170902270676\">Visit this<span>\u00a0<\/span><a href=\"https:\/\/openstax.org\/l\/21intquawavint\" rel=\"nofollow\">interactive simulation<\/a><span>\u00a0<\/span>to learn more about quantum wave interference.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1170903831619\">The square of the matter wave<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-9-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03a8|2<\/span><\/span><\/span><span>\u00a0<\/span>in one dimension has a similar interpretation as the square of the electric field<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-10-Frame\"><span class=\"MathJax_MathContainer\"><span>|E|2<\/span><\/span><\/span>. It gives the probability that a particle will be found at a particular position and time per unit length, also called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term282\">probability density<\/span>. The probability (<em data-effect=\"italics\">P<\/em>) a particle is found in a narrow interval (<em data-effect=\"italics\">x<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">x + dx<\/em>) at time<span>\u00a0<\/span><em data-effect=\"italics\">t<\/em><span>\u00a0<\/span>is therefore<\/p>\n<div data-type=\"equation\" id=\"fs-id1170904030971\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-11-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-11-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.1]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">(Later, we define the magnitude squared for the general case of a function with \u201cimaginary parts.\u201d) This probabilistic interpretation of the wave function is called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term283\" style=\"font-size: 14pt\">Born interpretation<\/span><span style=\"font-size: 14pt\">. Examples of wave functions and their squares for a particular time<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">t<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">are given in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.3.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Prob_Square\">\n<figure style=\"width: 486px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Three wave functions and the square of the amplitude of the wave functions. The first wave function, psi sub zero, and its square are symmetric, positive, peaked at x = 0, and zero far from the origin. The second wave function, psi sub 1, is antisymmetric: the function is negative at negative x, positive at positive x, and zero at the origin as well as plus and minus infinity. There is a negative minimum at negative x and positive maximum at positive x. The minimum value is exactly opposite the maximum value. The square of the amplitude of the wave function is positive and symmetric about the origin, where the value is zero, with a maximum on either side of the origin. The third wave function, psi sub N, is not symmetric. It is zero at minus infinity, decreases to a negative minimum value at some x less than zero, crosses zero, still at x less than zero, and becomes positive. It reaches a positive maximum at some positive x, then decreases to zero at large x. The minimum value is smaller in magnitude than the maximum value. The square of the amplitude of the wave function is positive, with two local maxima. The first maximum is smaller and at negative x, and the second is larger and at positive x. The square of the function is zero at one point between the maxima.\" data-media-type=\"image\/jpeg\" id=\"56881\" src=\"https:\/\/cnx.org\/resources\/2d838f6061b1af999544170bcc02c79688d82134\" width=\"486\" height=\"469\" \/><figcaption class=\"wp-caption-text\">Figure 3.3 Several examples of wave functions and the corresponding square of their wave functions.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170903805831\">If the wave function varies slowly over the interval<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-12-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0394x<\/span><\/span><\/span>, the probability a particle is found in the interval is approximately<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902089475\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-13-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-13-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248|\u03a8(x,t)|2\u0394x.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.2]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Notice that squaring the wave function ensures that the probability is positive. (This is analogous to squaring the electric field strength\u2014which may be positive or negative\u2014to obtain a positive value of intensity.) However, if the wave function does not vary slowly, we must integrate:<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170903900559\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-14-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-14-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)=\u222bxx+\u0394x|\u03a8(x,t)|2dx.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.3]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">This probability is just the area under the function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-15-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03a8(x,t)|2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><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-16-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">x+\u0394x<\/span><\/span><span style=\"font-size: 14pt\">. The probability of finding the particle \u201csomewhere\u201d (the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term284\" style=\"font-size: 14pt\">normalization condition<\/span><span style=\"font-size: 14pt\">) is<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-17-Frame\"><span class=\"MathJax_MathContainer\"><span>P(\u2212\u221e,+\u221e)=\u222b\u2212\u221e\u221e|\u03a8(x,t)|2dx=1.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.4]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">For a particle in two dimensions, the integration is over an area and requires a double integral; for a particle in three dimensions, the integration is over a volume and requires a triple integral. For now, we stick to the simple one-dimensional case.<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1170902178031\" class=\"ui-has-child-title\">\n<header><\/header>\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.1<\/span><span class=\"os-divider\"><\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170902089222\"><span data-type=\"title\"><strong>Where Is the Ball? (Part I)<\/strong><\/span><\/p>\n<p>A ball is constrained to move along a line inside a tube of length<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>. The ball is equally likely to be found anywhere in the tube at some time<span>\u00a0<\/span><em data-effect=\"italics\">t<\/em>. What is the probability of finding the ball in the left half of the tube at that time? (The answer is 50%, of course, but how do we get this answer by using the probabilistic interpretation of the quantum mechanical wave function?)<\/p>\n<p><span data-type=\"title\"><strong>Strategy<\/strong><\/span><\/p>\n<p>The first step is to write down the wave function. The ball is equally like to be found anywhere in the box, so one way to describe the ball with a<span>\u00a0<\/span><em data-effect=\"italics\">constant<\/em><span>\u00a0<\/span>wave function (Figure 3.4). The normalization condition can be used to find the value of the function and a simple integration over half of the box yields the final answer.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Square_Wave\">\n<figure style=\"width: 465px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The wave function Psi of x and t is plotted as a function of x. It is a step function, zero for x less than 0 and x greater than L, and constant for x between zero and L.\" data-media-type=\"image\/jpeg\" id=\"22284\" src=\"https:\/\/cnx.org\/resources\/1db9b1f532811d28b55a78a01f666df7b0461fef\" width=\"465\" height=\"284\" \/><figcaption class=\"wp-caption-text\">Figure 3.4 Wave function for a ball in a tube of length L.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><br \/>\n<strong>Solution<\/strong><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">The wave function of the ball can be written as<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-18-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=C(0&lt;x&lt;L),<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is a constant, and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-19-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03a8(x,t)=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">otherwise. We can determine the constant<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">by applying the normalization condition (we set<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-20-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">t=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to simplify the notation):<\/span><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902141124\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-21-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x=\u2212\u221e,+\u221e)=\u222b\u2212\u221e\u221e|C|2dx=1.<br \/>\n<\/span><span style=\"text-indent: 1em;font-size: 1rem\"><br \/>\nThis integral can be broken into three parts: (1) negative infinity to zero, (2) zero to<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">, and (3)<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to infinity. The particle is constrained to be in the tube, so<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-22-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">C=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">outside the tube and the first and last integrations are zero. The above equation can therefore be written<\/span><span><br \/>\n<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904208604\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-23-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>P(x=0,L)=\u222b0L|C|2dx=1.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The value<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">C<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">does not depend on<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and can be taken out of the integral, so we obtain<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903857790\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-24-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>|C|2\u222b0Ldx=1.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Integration gives<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902289643\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-25-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>C=1L.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">To determine the probability of finding the ball in the first half of the box<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-26-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(0&lt;x&lt;L),<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">we have<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904134011\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-27-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>P(x=0,L\/2)=\u222b0L\/2|1L|2dx=(1L)L2=0.50.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The probability of finding the ball in the first half of the tube is 50%, as expected. Two observations are noteworthy. First, this result corresponds to the area under the constant function from<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-28-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">to<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2 (the area of a square left of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2). Second, this calculation requires an integration of the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">square<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">of the wave function. A common mistake in performing such calculations is to forget to square the wave function before integration.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1170904209524\" class=\"ui-has-child-title\">\n<section>\n<div class=\"textbox shaded\">\n<div data-type=\"example\" id=\"fs-id1170902178031\" class=\"ui-has-child-title\">\n<section><span class=\"os-title-label\" style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">EXAMPLE 3<\/span><span class=\"os-number\" style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">.2<\/span><\/section>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1170904209524\" class=\"ui-has-child-title\">\n<header>\n<p class=\"os-title\"><strong>Where Is the Ball? (Part II)<\/strong><\/p>\n<p><span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">A ball is again constrained to move along a line inside a tube of length<\/span><span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">\u00a0<\/span><em style=\"font-family: Tinos, Georgia, serif;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">. This time, the ball is found preferentially in the middle of the tube. One way to represent its wave function is with a simple cosine function (<\/span>Figure 3.5<span style=\"font-family: Tinos, Georgia, serif;font-size: 1rem;font-style: normal\">). What is the probability of finding the ball in the last one-quarter of the tube?<\/span><br \/>\n<span class=\"os-divider\"><\/span><\/p>\n<\/header>\n<section>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Cosine_Wave\">\n<figure style=\"width: 426px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"A plot of Psi of x at t equal to zero as a function of x. The function is zero for x less than minus L over 2 and x greater than L over 2. For x between minus and plus L over 2, the functions is a cosine curve, concave down with a positive maximum at x equal to zero, and going to zero at minus and plus L over 2.\" data-media-type=\"image\/jpeg\" id=\"93531\" src=\"https:\/\/cnx.org\/resources\/07ab66f742ac10c60f8b3ae4bb8c4efcfb52521a\" width=\"426\" height=\"295\" \/><figcaption class=\"wp-caption-text\">Figure 3.5 Wave function for a ball in a tube of length L, where the ball is preferentially in the middle of the tube.<span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><\/span><\/span><\/span><br \/><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><\/span><\/span><\/span><\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\"><span class=\"os-caption\"><span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><br \/>\n<strong>Strategy<\/strong><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"os-caption\"><span class=\"os-caption\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><\/span><span style=\"text-indent: 1em;font-size: 1rem\">We use the same strategy as before. In this case, the wave function has two unknown constants: One is associated with the wavelength of the wave and the other is the amplitude of the wave. We determine the amplitude by using the boundary conditions of the problem, and we evaluate the wavelength by using the normalization condition. Integration of the square of the wave function over the last quarter of the tube yields the final answer. The calculation is simplified by centering our coordinate system on the peak of the wave function.<\/p>\n<p><\/span><\/span><\/span><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span><\/span><\/span><\/p>\n<p><span style=\"font-size: 1rem;text-indent: 1em\">The wave function of the ball can be written<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903814492\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-29-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03a8(x,0)=Acos(kx)(\u2212L\/2&lt;x&lt;L\/2),<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">A<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is the amplitude of the wave function and<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-30-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">k=2\u03c0\/\u03bb<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">is its wave number. Beyond this interval, the amplitude of the wave function is zero because the ball is confined to the tube. Requiring the wave function to terminate at the right end of the tube gives<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902216824\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-31-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03a8(x=L2,0)=0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Evaluating the wave function at<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-32-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L\/2<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">gives<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903906927\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-33-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>Acos(kL\/2)=0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">This equation is satisfied if the argument of the cosine is an integral multiple of<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-34-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c0\/2,3\u03c0\/2,5\u03c0\/2,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">and so on. In this case, we have<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902132492\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-35-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>kL2=\u03c02,<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">or<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903756446\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-36-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>k=\u03c0L.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Applying the normalization condition gives<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-37-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">A=2\/L<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, so the wave function of the ball is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170904271710\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-38-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03a8(x,0)=2Lcos(\u03c0x\/L),\u2212L\/2&lt;x&lt;L\/2.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">To determine the probability of finding the ball in the last quarter of the tube, we square the function and integrate:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902339022\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-39-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>P(x=L\/4,L\/2)=\u222bL\/4L\/2|2Lcos(\u03c0xL)|2dx=0.091.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The probability of finding the ball in the last quarter of the tube is 9.1%. The ball has a definite wavelength<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-40-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(\u03bb=2L)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. If the tube is of macroscopic length<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-41-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(L=1m)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">, the momentum of the ball is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902275088\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-42-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>p=h\u03bb=h2L~10\u221236m\/s.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">This momentum is much too small to be measured by any human instrument.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p><span style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">An Interpretation of the Wave Function<\/span><\/p>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1170902128957\" data-depth=\"1\">\n<p id=\"fs-id1170904168306\">We are now in position to begin to answer the questions posed at the beginning of this section. First, for a traveling particle described by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-43-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Asin(kx\u2212\u03c9t)<\/span><\/span><\/span>, what is \u201cwaving?\u201d Based on the above discussion, the answer is a mathematical function that can, among other things, be used to determine where the particle is likely to be when a position measurement is performed. Second, how is the wave function used to make predictions? If it is necessary to find the probability that a particle will be found in a certain interval, square the wave function and integrate over the interval of interest. Soon, you will learn soon that the wave function can be used to make many other kinds of predictions, as well.<\/p>\n<p id=\"fs-id1170904071853\">Third, if a matter wave is given by the wave function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-44-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)<\/span><\/span><\/span>,<span>\u00a0<\/span><em data-effect=\"italics\">where<\/em><span>\u00a0<\/span>exactly is the particle? Two answers exist: (1) when the observer<span>\u00a0<\/span><em data-effect=\"italics\">is not<\/em><span>\u00a0<\/span>looking (or the particle is not being otherwise detected), the particle is everywhere<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-45-Frame\"><span class=\"MathJax_MathContainer\"><span>(x=\u2212\u221e,+\u221e)<\/span><\/span><\/span>; and (2) when the observer<span>\u00a0<\/span><em data-effect=\"italics\">is<\/em><span>\u00a0<\/span>looking (the particle is being detected), the particle \u201cjumps into\u201d a particular position state<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-46-Frame\"><span class=\"MathJax_MathContainer\"><span>(x,x+dx)<\/span><\/span><\/span><span>\u00a0<\/span>with a probability given by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-47-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx<\/span><\/span><\/span><span>\u00a0<\/span>\u2014a process called<span>\u00a0<\/span><span data-type=\"term\" id=\"term285\">state reduction<\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span data-type=\"term\" id=\"term286\">wave function collapse<\/span>. This answer is called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term287\">Copenhagen interpretation<\/span><span>\u00a0<\/span>of the wave function, or of quantum mechanics.<\/p>\n<p id=\"fs-id1170902247796\">To illustrate this interpretation, consider the simple case of a particle that can occupy a small container either at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-48-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-49-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>(Figure 3.6). In classical physics, we assume the particle is located either at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-50-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-51-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>when the observer is not looking. However, in quantum mechanics, the particle may exist in a state of indefinite position\u2014that is, it may be located at<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-52-Frame\"><span class=\"MathJax_MathContainer\"><span>x1<\/span><\/span><\/span><span>\u00a0<\/span><em data-effect=\"italics\">and<\/em><span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-53-Frame\"><span class=\"MathJax_MathContainer\"><span>x2<\/span><\/span><\/span><span>\u00a0<\/span>when the observer is not looking. The assumption that a particle can only have one value of position (when the observer is not looking) is abandoned. Similar comments can be made of other measurable quantities, such as momentum and energy.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Two_State\">\n<figure style=\"width: 500px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"An x y coordinate system is shown with two small boxes drawn on the x axis, one at x sub 1 to the left of the origin and the other at x sub 2 to the right of the origin.\" data-media-type=\"image\/jpeg\" id=\"73184\" src=\"https:\/\/cnx.org\/resources\/139a50ec55ffda7dd8f33fe25a5b172e28353f54\" width=\"500\" height=\"172\" \/><figcaption class=\"wp-caption-text\">Figure 3.6 A two-state system of position of a particle.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170903864386\">The bizarre consequences of the Copenhagen interpretation of quantum mechanics are illustrated by a creative thought experiment first articulated by Erwin Schr\u00f6dinger (<em data-effect=\"italics\">National Geographic<\/em>, 2013) (Figure 3.7):<\/p>\n<p id=\"fs-id1170903875163\">\u201cA cat is placed in a steel box along with a Geiger counter, a vial of poison, a hammer, and a radioactive substance. When the radioactive substance decays, the Geiger detects it and triggers the hammer to release the poison, which subsequently kills the cat. The radioactive decay is a random [probabilistic] process, and there is no way to predict when it will happen. Physicists say the atom exists in a state known as a superposition\u2014both decayed and not decayed at the same time. Until the box is opened, an observer doesn\u2019t know whether the cat is alive or dead\u2014because the cat\u2019s fate is intrinsically tied to whether or not the atom has decayed and the cat would [according to the Copenhagen interpretation] be \u201cliving and dead &#8230; in equal parts\u201d until it is observed.\u201d<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Cat\">\n<figure style=\"width: 974px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The Schodinger cat thought experiment, consisting of a Geiger counter, a vial of poison, a hammer, a radioactive substance, and a cat is illustrated. Each is shown in two states: The Geiger counter triggered and untriggered, the hammer up and down, the poison vial whole and broken, and the cat alive and dead.\" data-media-type=\"image\/jpeg\" id=\"93175\" src=\"https:\/\/cnx.org\/resources\/ba43497f5351605d6b1cf05c2744d5852ede67ca\" width=\"974\" height=\"518\" \/><figcaption class=\"wp-caption-text\">Figure 3.7 Schr\u00f6dinger\u2019s cat. Schr\u00f6dinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170903874503\">Two-state systems (left and right, atom decays and does not decay, and so on) are often used to illustrate the principles of quantum mechanics. These systems find many applications in nature, including electron spin and mixed states of particles, atoms, and even molecules. Two-state systems are also finding application in the quantum computer, as mentioned in the introduction of this chapter. Unlike a digital computer, which encodes information in binary digits (zeroes and ones), a quantum computer stores and manipulates data in the form of quantum bits, or qubits. In general, a<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term289\">qubit<\/span><span>\u00a0<\/span>is not in a state of zero or one, but rather in a mixed state of zero<span>\u00a0<\/span><em data-effect=\"italics\">and<\/em>one. If a large number of qubits are placed in the same quantum state, the measurement of an individual qubit would produce a zero with a probability<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>, and a one with a probability<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-54-Frame\"><span class=\"MathJax_MathContainer\"><span>q=1\u2212p.<\/span><\/span><\/span><span>\u00a0<\/span>Many scientists believe that quantum computers are the future of the computer industry.<\/p>\n<\/section>\n<section id=\"fs-id1170902337169\" data-depth=\"1\">\n<h3 data-type=\"title\">Complex Conjugates<\/h3>\n<p id=\"fs-id1170903874497\">Later in this section, you will see how to use the wave function to describe particles that are \u201cfree\u201d or bound by forces to other particles. The specific form of the wave function depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually\u00a0<span data-type=\"term\" id=\"term290\">complex function<\/span><strong data-effect=\"bold\">s<\/strong>. A complex function is one that contains one or more imaginary numbers\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-55-Frame\"><span class=\"MathJax_MathContainer\"><span>(i=\u22121)<\/span><\/span><\/span>. Experimental measurements produce real (nonimaginary) numbers only, so the above procedure to use the wave function must be slightly modified. In general, the probability that a particle is found in the narrow interval (<em data-effect=\"italics\">x<\/em>,<em data-effect=\"italics\">x + dx<\/em>) at time<em data-effect=\"italics\">t<\/em>is given by<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-56-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+dx)=|\u03a8(x,t)|2dx=\u03a8*(x,t)\u03a8(x,t)dx,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.5]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902243906\">where<span class=\"MathJax_MathML\" id=\"MathJax-Element-57-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)<\/span><\/span><\/span>is the complex conjugate of the wave function. The complex conjugate of a function is obtaining by replacing every occurrence of<span class=\"MathJax_MathML\" id=\"MathJax-Element-58-Frame\"><span class=\"MathJax_MathContainer\"><span>i=\u22121<\/span><\/span><\/span>in that function with<span class=\"MathJax_MathML\" id=\"MathJax-Element-59-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212i<\/span><\/span><\/span>. This procedure eliminates complex numbers in all predictions because the product<span class=\"MathJax_MathML\" id=\"MathJax-Element-60-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)\u03a8(x,t)<\/span><\/span><\/span>is always a real number.<\/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\">.1<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header><span style=\"font-size: 1rem\">If<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-61-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">a=3+4i<\/span><\/span><span style=\"font-size: 1rem\">, what is the product<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-62-Frame\" style=\"font-size: 1rem\"><span class=\"MathJax_MathContainer\">a*a<\/span><\/span><span style=\"font-size: 1rem\">?<\/span><\/header>\n<\/div>\n<\/div>\n<p id=\"fs-id1170903874962\">Consider the motion of a free particle that moves along the\u00a0<em data-effect=\"italics\">x<\/em>-direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wave function has real\u00a0<em data-effect=\"italics\">and\u00a0<\/em>complex parts. In particular, the wave function is given by<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903886955\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-63-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Acos(kx\u2212\u03c9t)+iAsin(kx\u2212\u03c9t),<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170904060990\">where\u00a0<em data-effect=\"italics\">A\u00a0<\/em>is the amplitude,\u00a0<em data-effect=\"italics\">k\u00a0<\/em>is the wave number, and<span class=\"MathJax_MathML\" id=\"MathJax-Element-64-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c9<\/span><\/span><\/span>is the angular frequency. Using Euler\u2019s formula,<span class=\"MathJax_MathML\" id=\"MathJax-Element-65-Frame\"><span class=\"MathJax_MathContainer\"><span>ei\u03d5=cos(\u03d5)+isin(\u03d5),<\/span><\/span><\/span>this equation can be written in the form<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902187505\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-66-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t)=Aei(kx\u2212\u03c9t)=Aei\u03d5,<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902097748\">where<span class=\"MathJax_MathML\" id=\"MathJax-Element-67-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span>is the phase angle. If the wave function varies slowly over the interval<span class=\"MathJax_MathML\" id=\"MathJax-Element-68-Frame\"><span class=\"MathJax_MathContainer\"><span>\u0394x,<\/span><\/span><\/span>the probability of finding the particle in that interval is<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902214038\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-69-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248\u03a8*(x,t)\u03a8(x,t)\u0394x=(Aei\u03d5)(A*e\u2212i\u03d5)\u0394x=(A*A)\u0394x.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902041507\">If<em data-effect=\"italics\">A<\/em>has real and complex parts<span class=\"MathJax_MathML\" id=\"MathJax-Element-70-Frame\"><span class=\"MathJax_MathContainer\"><span>(a+ib<\/span><\/span><\/span>, where\u00a0<em data-effect=\"italics\">a\u00a0<\/em>and\u00a0<em data-effect=\"italics\">b\u00a0<\/em>are real constants), then<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903751129\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-71-Frame\"><span class=\"MathJax_MathContainer\"><span>A*A=(a+ib)(a\u2212ib)=a2+b2.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902091680\">Notice that the complex numbers have vanished. Thus,<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902300451\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-72-Frame\"><span class=\"MathJax_MathContainer\"><span>P(x,x+\u0394x)\u2248|A|2\u0394x<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170903832048\">is a real quantity. The interpretation of<span class=\"MathJax_MathML\" id=\"MathJax-Element-73-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8*(x,t)\u03a8(x,t)<\/span><\/span><\/span>as a probability density ensures that the predictions of quantum mechanics can be checked in the \u201creal world.\u201d<\/p>\n<\/section>\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\">.2<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header><span style=\"font-size: 1rem\">Suppose that a particle with energy<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">E<\/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 and is confined in the region between 0 and<\/span><span style=\"font-size: 1rem\">\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">. One possible wave function is<\/span><\/header>\n<section>\n<div class=\"os-note-body\">\n<div class=\"os-hasSolution unnumbered\" data-type=\"exercise\" id=\"fs-id1170904134843\">\n<section>\n<div data-type=\"problem\" id=\"fs-id1170902114584\">\n<div class=\"os-problem-container\">\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1172100931629\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-74-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x,t)={Ae\u2212iEt\/\u210fsin\u03c0xL,when 0\u2264x\u2264L0,otherwise.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902189634\">Determine the normalization constant.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p><span style=\"font-family: Roboto, Helvetica, Arial, sans-serif;font-size: 1em;font-style: italic\">Expectation Values<\/span><\/p>\n<section id=\"fs-id1170903851901\" data-depth=\"1\">\n<p id=\"fs-id1170903874481\">In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as <em data-effect=\"italics\">x<\/em>(<em data-effect=\"italics\">t<\/em>), where\u00a0<em data-effect=\"italics\">x\u00a0<\/em>is the position and\u00a0<em data-effect=\"italics\">t\u00a0<\/em>is the time. Note that the particle has one value of position for any time <em data-effect=\"italics\">t<\/em>. In quantum mechanics, however, the solution to an equation of motion is a wave function,<span class=\"MathJax_MathML\" id=\"MathJax-Element-75-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03a8(x,t). <\/span><\/span><\/span>The particle has many values of position for any time <em data-effect=\"italics\">t<\/em>, and only the probability density of finding the particle,<span class=\"MathJax_MathML\" id=\"MathJax-Element-76-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03a8(x,t)|2<\/span><\/span><\/span>, can be known. The average value of position for a large number of particles with the same wave function is expected to be<\/p>\n<div data-type=\"equation\" id=\"fs-id1170904189252\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-77-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-77-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e\u221exP(x,t)dx=\u222b\u2212\u221e\u221ex\u03a8*(x,t)\u03a8(x,t)dx.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.6]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">This is called the\u00a0<\/span><span data-type=\"term\" id=\"term291\" style=\"font-size: 14pt\">expectation value o<\/span><span style=\"font-size: 14pt\">f the position. It is usually written<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-78-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e\u221e\u03a8*(x,t)x\u03a8(x,t)dx,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.7]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where the <\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x <\/em><span style=\"font-size: 14pt\">is sandwiched between the wave functions. The reason for this will become apparent soon. Formally,<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">is called the\u00a0<\/span><span data-type=\"term\" id=\"term292\" style=\"font-size: 14pt\">position operator<\/span><span style=\"font-size: 14pt\">.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902235000\">At this point, it is important to stress that a wave function can be written in terms of other quantities as well, such as velocity (<em data-effect=\"italics\">v<\/em>), momentum (<em data-effect=\"italics\">p<\/em>), and kinetic energy (<em data-effect=\"italics\">K<\/em>). The expectation value of momentum, for example, can be written<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902149970\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-79-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-79-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b\u2212\u221e\u221e\u03a8*(p,t)p\u03a8(p,t)dp,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.8]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Where\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dp\u00a0<\/em><span style=\"font-size: 14pt\">is used instead of\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">dx\u00a0<\/em><span style=\"font-size: 14pt\">to indicate an infinitesimal interval in momentum. In some cases, we know the wave function in position,<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-80-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03a8(x,t),<\/span><\/span><span style=\"font-size: 14pt\">but seek the expectation of momentum. The procedure for doing this is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902208206\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-81-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-81-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b\u2212\u221e\u221e\u03a8*(x,t)(\u2212i\u210fddx)\u03a8(x,t)dx,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.9]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where the quantity in parentheses, sandwiched between the wave functions, is called the\u00a0<\/span><span data-type=\"term\" id=\"term293\" style=\"font-size: 14pt\">momentum operator\u00a0<\/span><span style=\"font-size: 14pt\">in the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-direction. [The momentum operator in Equation 3.9is said to be the position-space representation of the momentum operator.] The momentum operator must act (operate) on the wave function to the right, and then the result must be multiplied by the complex conjugate of the wave function on the left, before integration. The momentum operator in the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-direction is sometimes denoted<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170904245770\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-82-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-82-Frame\"><span class=\"MathJax_MathContainer\"><span>(px)op=\u2212i\u210fddx,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.10]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Momentum operators for the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">&#8211; and <\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-directions are defined similarly. This operator and many others are derived in a more advanced course in modern physics. In some cases, this derivation is relatively simple. For example, the kinetic energy operator is just<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902293433\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-83-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-83-Frame\"><span class=\"MathJax_MathContainer\"><span>(K)op=12m(vx)op2=(px)op22m=(\u2212i\u210fddx)22m=\u2212\u210f22m(ddx)(ddx).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.11]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Thus, if we seek an expectation value of kinetic energy of a particle in one dimension, two successive ordinary derivatives of the wave function are required before integration.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902271392\">Expectation-value calculations are often simplified by exploiting the symmetry of wave functions. Symmetric wave functions can be even or odd. An\u00a0<span data-type=\"term\" id=\"term294\">even function\u00a0<\/span>is a function that satisfies<\/p>\n<div data-type=\"equation\" id=\"eip-240\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-84-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-84-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)=\u03c8(\u2212x).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.12]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In contrast, an\u00a0<\/span><span data-type=\"term\" id=\"term295\" style=\"font-size: 14pt\">odd function\u00a0<\/span><span style=\"font-size: 14pt\">is a function that satisfies<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"eip-912\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-85-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-85-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8(x)=-\u03c8(\u2212x).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.13]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">An example of even and odd functions is shown in Figure 3.8. An even function is symmetric about the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis. This function is produced by reflecting<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-86-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">for<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">&gt; 0 about the vertical\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis. By comparison, an odd function is generated by reflecting the function about the<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt\">-axis and then about the\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">-axis. (An odd function is also referred to as an\u00a0<\/span><span data-type=\"term\" id=\"term296\" style=\"font-size: 14pt\">anti-symmetric function<\/span><span style=\"font-size: 14pt\">.)<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_01_Even_Odd\">\n<figure style=\"width: 381px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Two wave functions are plotted as a function of x. The vertical scale runs form -0.5 to +0.5 and the horizontal scale from -4 to 4. The even function is plotted in blue. It is symmetric about the origin, positive for all values of x, and going to zero at the ends. This particular even function has a positive minimum at the origin and maxima on either side. The odd function is zero at the origin and at the ends, negative to the left of the origin, where it has a minimum, and positive to the right, where it has a maximum. The function is antisymmetric, meaning that the negative half is the same shape as the right half, but inverted, that is, generated by reflecting the function about the y-axis and then about the x-axis.\" data-media-type=\"image\/jpeg\" id=\"99542\" src=\"https:\/\/cnx.org\/resources\/e6dca3667a707a47cd782cc5794d7bbfce297779\" width=\"381\" height=\"233\" \/><figcaption class=\"wp-caption-text\">Figure 3.8 Examples of even and odd wave functions.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170902236504\">In general, an even function times an even function produces an even function. A simple example of an even function is the product\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-87-Frame\"><span class=\"MathJax_MathContainer\"><span>x2e\u2212x2\u00a0<\/span><\/span><\/span>(even times even is even). Similarly, an odd function times an odd function produces an even function, such as\u00a0<em data-effect=\"italics\">x<\/em>s in\u00a0<em data-effect=\"italics\">x\u00a0<\/em>(odd times odd is even). However, an odd function times an even function produces an odd function, such as\u00a0<span class=\"MathJax_MathML\" id=\"MathJax-Element-88-Frame\"><span class=\"MathJax_MathContainer\"><span>xe\u2212x2<\/span><\/span><\/span>(odd times even is odd). The integral over all space of an odd function is zero, because the total area of the function above the\u00a0<em data-effect=\"italics\">x<\/em>-axis cancels the (negative) area below it. As the next example shows, this property of odd functions is very useful.<\/p>\n<div data-type=\"example\" id=\"fs-id1170904072147\" class=\"ui-has-child-title\">\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.3<\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170902178244\"><span data-type=\"title\"><strong>Expectation Value (Part I)<\/strong><\/span><\/p>\n<p>The normalized wave function of a particle is<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902178250\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-89-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03c8(x)=e\u2212|x|\/x0\/x0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Find the expectation value of position.<\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Substitute the wave function into\u00a0<\/span>Equation 3.7<span style=\"text-indent: 1em;font-size: 1rem\">and evaluate. The position operator introduces a multiplicative factor only, so the position operator need not be \u201csandwiched.\u201d<\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">First multiply, then integrate:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170900367549\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-90-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b\u2212\u221e+\u221edxx|\u03c8(x)|2=\u222b\u2212\u221e+\u221edxx|e\u2212|x|\/x0x0|2=1&#215;0\u222b\u2212\u221e+\u221edxxe\u22122|x|\/x0=0.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The function in the integrand<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-91-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\"> (xe\u22122|x|\/x0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is odd since it is the product of an odd function (<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">) and an even function<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-92-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(e\u22122|x|\/x0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. The integral vanishes because the total area of the function about the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-axis cancels the (negative) area below it. The result<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-93-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">(\u2329x\u232a=0)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is not surprising since the probability density function is symmetric about\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-94-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.4<\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170902346418\"><span data-type=\"title\"><strong>Expectation Value (Part II)<\/strong><\/span><\/p>\n<p>The time-dependent wave function of a particle confined to a region between 0 and <em data-effect=\"italics\">L <\/em>is<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902271963\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-95-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03c8(x,t)=Ae\u2212i\u03c9tsin(\u03c0x\/L)<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">where<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-96-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c9<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">is angular frequency and <\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">E <\/em><span style=\"text-indent: 1em;font-size: 1rem\">is the energy of the particle. (<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">Note:<\/em><span style=\"text-indent: 1em;font-size: 1rem\">The function varies as a sine because of the limits (0 to<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">). When\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-97-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0,<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">the sine factor is zero and the wave function is zero, consistent with the boundary conditions.) Calculate the expectation values of position, momentum, and kinetic energy.<\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Strategy<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">We must first normalize the wave function to find\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">A<\/em><span style=\"text-indent: 1em;font-size: 1rem\">. Then we use the operators to calculate the expectation values.<\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Solution<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Computation of the normalization constant:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902272294\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-98-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>1=\u222b0Ldx\u03c8*(x)\u03c8(x)=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(Ae\u2212i\u03c9tsin\u03c0xL)=A2\u222b0Ldxsin2\u03c0xL=A2L2\u21d2A=2L.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of position is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902203736\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-99-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2329x\u232a=\u222b0Ldx\u03c8*(x)x\u03c8(x)=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)x(Ae\u2212i\u03c9tsin\u03c0xL)=A2\u222b0Ldxxsin2\u03c0xL=A2L24=L2.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of momentum in the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction also requires an integral. To set this integral up, the associated operator must\u2014 by rule\u2014act to the right on the wave function<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-100-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902344048\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-101-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2212i\u210fddx\u03c8(x)=\u2212i\u210fddxAe\u2212i\u03c9tsin\u03c0xL=\u2212iAh2Le\u2212i\u03c9tcos\u03c0xL.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Therefore, the expectation value of momentum is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902266047\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-102-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2329p\u232a=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(\u2212iAh2Le\u2212i\u03c9tcos\u03c0xL)=\u2212iA2h4L\u222b0Ldxsin2\u03c0xL=0.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The function in the integral is a sine function with a wavelength equal to the width of the well,<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u2014an odd function about\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-103-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L\/2<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">. As a result, the integral vanishes.<\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The expectation value of kinetic energy in the\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">x<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-direction requires the associated operator to act on the wave function:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902110544\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-104-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2dx2\u03c8(x)=\u2212\u210f22md2dx2Ae\u2212i\u03c9tsin\u03c0xL=\u2212\u210f22mAe\u2212i\u03c9td2dx2sin\u03c0xL=Ah22mL2e\u2212i\u03c9tsin\u03c0xL.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">Thus, the expectation value of the kinetic energy is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902062116\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-105-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u2329K\u232a=\u222b0Ldx(Ae+i\u03c9tsin\u03c0xL)(Ah22mL2e\u2212i\u03c9tsin\u03c0xL)=A2h22mL2\u222b0Ldxsin2\u03c0xL=A2h22mL2L2=h22mL2.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">The average position of a large number of particles in this state is<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2. The average momentum of these particles is zero because a given particle is equally likely to be moving right or left. However, the particle is not at rest because its average kinetic energy is not zero. Finally, the probability density is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902255479\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-106-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>|\u03c8|2=(2\/L)sin2(\u03c0x\/L).<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">This probability density is largest at location\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\/2 and is zero at<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-107-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=0<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">and at<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-108-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">x=L.<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">Note that these conclusions do not depend explicitly on time.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\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\">.3<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header>\n<div class=\"os-title\"><span style=\"font-size: 1rem\">For the particle in the above example, find the probability of locating it between positions 0 and\u00a0<\/span><em style=\"font-size: 1rem\" data-effect=\"italics\">L<\/em><span style=\"font-size: 1rem\">\/4<\/span><\/div>\n<\/header>\n<\/div>\n<\/div>\n<p id=\"fs-id1170904139295\">Quantum mechanics makes many surprising predictions. However, in 1920, Niels Bohr (founder of the Niels Bohr Institute in Copenhagen, from which we get the term \u201cCopenhagen interpretation\u201d) asserted that the predictions of quantum mechanics and classical mechanics must agree for all macroscopic systems, such as orbiting planets, bouncing balls, rocking chairs, and springs. This\u00a0<span data-type=\"term\" id=\"term297\">correspondence principle\u00a0<\/span>is now generally accepted. It suggests the rules of classical mechanics are an approximation of the rules of quantum mechanics for systems with very large energies. Quantum mechanics describes both the microscopic and macroscopic world, but classical mechanics describes only the latter.<\/p>\n<\/section>\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<\/section>\n","protected":false},"author":615,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"3. Quantum Mechanics","pb_subtitle":"3.1 Wave Functions","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-187","chapter","type-chapter","status-publish","hentry"],"part":179,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/187","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/users\/615"}],"version-history":[{"count":9,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/187\/revisions"}],"predecessor-version":[{"id":476,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/187\/revisions\/476"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/parts\/179"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/187\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/media?parent=187"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapter-type?post=187"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/contributor?post=187"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/license?post=187"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}