Modern Physics: Quantum Mechanics (PDF)

bei48482_ch05.qxd 1/17/02 12:17 AM Page 160 CHAPTER 5 Quantum Mechanics Scanning tunneling micrograph of gold atoms on a carbon (graphite) substrate....

bei48482_ch05.qxd 1/17/02 12:17 AM Page 160 CHAPTER 5 Quantum Mechanics Scanning tunneling micrograph of gold atoms on a carbon (graphite) substrate. The cluster of gold atoms is about 1.5 nm across and three atoms high. 5.1 QUANTUM MECHANICS 5.7 SCHRÖDINGER’S EQUATION: Classical mechanics is an approximation of STEADY-STATE FORM quantum mechanics Eigenvalues and eigenfunctions 5.2 THE WAVE EQUATION 5.8 PARTICLE IN A BOX It can have a variety of solutions, including How boundary conditions and normalization complex ones determine wave functions 5.3 SCHRÖDINGER’S EQUATION: 5.9 FINITE POTENTIAL WELL TIME-DEPENDENT FORM The wave function penetrates the walls, which A basic physical principle that cannot be derived lowers the energy levels from anything else 5.10 TUNNEL EFFECT 5.4 LINEARITY AND SUPERPOSITION A particle without the energy to pass over a Wave functions add, not probabilities potential barrier may still tunnel through it 5.5 EXPECTATION VALUES 5.11 HARMONIC OSCILLATOR How to extract information from a wave Its energy levels are evenly spaced function APPENDIX: THE TUNNEL EFFECT 5.6 OPERATORS Another way to find expectation values 160 bei48482_ch05.qxd 2/4/02 11:37 AM Page 161 Quantum Mechanics 161 lthough the Bohr theory of the atom, which can be extended further than was A done in Chap. 4, is able to account for many aspects of atomic phenomena, it has a number of severe limitations as well. First of all, it applies only to hy- drogen and one-electron ions such as He and Li2—it does not even work for ordinary helium. The Bohr theory cannot explain why certain spectral lines are more intense than others (that is, why certain transitions between energy levels have greater probabilities of occurrence than others). It cannot account for the observation that many spectral lines actually consist of several separate lines whose wavelengths differ slightly. And perhaps most important, it does not permit us to obtain what a really suc- cessful theory of the atom should make possible: an understanding of how individual atoms interact with one another to endow macroscopic aggregates of matter with the physical and chemical properties we observe. The preceding objections to the Bohr theory are not put forward in an unfriendly way, for the theory was one of those seminal achievements that transform scientific thought, but rather to emphasize that a more general approach to atomic phenomena is required. Such an approach was developed in 1925 and 1926 by Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac, and others under the apt name of quantum mechanics. “The discovery of quantum mechanics was nearly a total surprise. It de- scribed the physical world in a way that was fundamentally new. It seemed to many of us a miracle,” noted Eugene Wigner, one of the early workers in the field. By the early 1930s the application of quantum mechanics to problems involving nuclei, atoms, molecules, and matter in the solid state made it possible to understand a vast body of data (“a large part of physics and the whole of chemistry,” according to Dirac) and— vital for any theory—led to predictions of remarkable accuracy. Quantum mechanics has survived every experimental test thus far of even its most unexpected conclusions. 5.1 QUANTUM MECHANICS Classical mechanics is an approximation of quantum mechanics The fundamental difference between classical (or Newtonian) mechanics and quantum mechanics lies in what they describe. In classical mechanics, the future history of a par- ticle is completely determined by its initial position and momentum together with the forces that act upon it. In the everyday world these quantities can all be determined well enough for the predictions of Newtonian mechanics to agree with what we find. Quantum mechanics also arrives at relationships between observable quantities, but the uncertainty principle suggests that the nature of an observable quantity is differ- ent in the atomic realm. Cause and effect are still related in quantum mechanics, but what they concern needs careful interpretation. In quantum mechanics the kind of cer- tainty about the future characteristic of classical mechanics is impossible because the initial state of a particle cannot be established with sufficient accuracy. As we saw in Sec. 3.7, the more we know about the position of a particle now, the less we know about its momentum and hence about its position later. The quantities whose relationships quantum mechanics explores are probabilities. Instead of asserting, for example, that the radius of the electron’s orbit in a ground- state hydrogen atom is always exactly 5.3  1011 m, as the Bohr theory does, quantum mechanics states that this is the most probable radius. In a suitable experiment most trials will yield a different value, either larger or smaller, but the value most likely to be found will be 5.3  1011 m. bei48482_ch05.qxd 1/17/02 12:17 AM Page 162 162 Chapter Five Quantum mechanics might seem a poor substitute for classical mechanics. However, classical mechanics turns out to be just an approximate version of quantum mechanics. The certainties of classical mechanics are illusory, and their apparent agreement with experiment occurs because ordinary objects consist of so many individual atoms that departures from average behavior are unnoticeable. Instead of two sets of physical prin- ciples, one for the macroworld and one for the microworld, there is only the single set included in quantum mechanics. Wave Function As mentioned in Chap. 3, the quantity with which quantum mechanics is concerned is the wave function  of a body. While  itself has no physical interpretation, the square of its absolute magnitude 2 evaluated at a particular place at a particular time is proportional to the probability of finding the body there at that time. The linear mo- mentum, angular momentum, and energy of the body are other quantities that can be established from . The problem of quantum mechanics is to determine  for a body when its freedom of motion is limited by the action of external forces. Wave functions are usually complex with both real and imaginary parts. A proba- bility, however, must be a positive real quantity. The probability density 2 for a com- plex  is therefore taken as the product * of  and its complex conjugate *. The complex conjugate of any function is obtained by replacing i (1 ) by i wherever it appears in the function. Every complex function  can be written in the form Wave function   A  iB where A and B are real functions. The complex conjugate * of  is Complex conjugate *  A  iB and so 2  *  A2  i2B2  A2  B2 since i2  1. Hence 2  * is always a positive real quantity, as required. Normalization Even before we consider the actual calculation of , we can establish certain require- ments it must always fulfill. For one thing, since 2 is proportional to the probabil- ity density P of finding the body described by , the integral of 2 over all space must be finite—the body is somewhere, after all. If   2 dV  0 the particle does not exist, and the integral obviously cannot be  and still mean any- thing. Furthermore, 2 cannot be negative or complex because of the way it is de- fined. The only possibility left is that the integral be a finite quantity if  is to describe properly a real body. It is usually convenient to have 2 be equal to the probability density P of find- ing the particle described by , rather than merely be proportional to P. If 2 is to bei48482_ch05.qxd 1/17/02 12:17 AM Page 163 Quantum Mechanics 163 equal P, then it must be true that Normalization   2 dV  1 (5.1) since if the particle exists somewhere at all times,    P dV  1 A wave function that obeys Eq. (5.1) is said to be normalized. Every acceptable wave function can be normalized by multiplying it by an appropriate constant; we shall shortly see how this is done. Well-Behaved Wave Functions Besides being normalizable,  must be single-valued, since P can have only one value at a particular place and time, and continuous. Momentum considerations (see Sec. 5.6) require that the partial derivatives x, y, z be finite, continuous, and single- valued. Only wave functions with all these properties can yield physically meaningful results when used in calculations, so only such “well-behaved” wave functions are ad- missible as mathematical representations of real bodies. To summarize: 1  must be continuous and single-valued everywhere. 2 x, y, z must be continuous and single-valued everywhere. 3  must be normalizable, which means that  must go to 0 as x → , y → , z →  in order that  2 dV over all space be a finite constant. These rules are not always obeyed by the wave functions of particles in model situations that only approximate actual ones. For instance, the wave functions of a par- ticle in a box with infinitely hard walls do not have continuous derivatives at the walls, since   0 outside the box (see Fig. 5.4). But in the real world, where walls are never infinitely hard, there is no sharp change in  at the walls (see Fig. 5.7) and the de- rivatives are continuous. Exercise 7 gives another example of a wave function that is not well-behaved. Given a normalized and otherwise acceptable wave function , the probability that the particle it describes will be found in a certain region is simply the integral of the probability density 2 over that region. Thus for a particle restricted to motion in the x direction, the probability of finding it between x1 and x2 is given by Probability Px1x2  x2 x1 2 dx (5.2) We will see examples of such calculations later in this chapter and in Chap. 6. 5.2 THE WAVE EQUATION It can have a variety of solutions, including complex ones Schrödinger’s equation, which is the fundamental equation of quantum mechanics in the same sense that the second law of motion is the fundamental equation of New- tonian mechanics, is a wave equation in the variable . bei48482_ch05.qxd 1/17/02 12:17 AM Page 164 164 Chapter Five Before we tackle Schrödinger’s equation, let us review the wave equation 2y 1 2y Wave equation  (5.3) x2 2 t2 which governs a wave whose variable quantity is y that propagates in the x direction with the speed . In the case of a wave in a stretched string, y is the displacement of the string from the x axis; in the case of a sound wave, y is the pressure difference; in the case of a light wave, y is either the electric or the magnetic field magnitude. Equation (5.3) can be derived from the second law of motion for mechanical waves and from Maxwell’s equations for electromagnetic waves. Partial Derivatives S uppose we have a function f(x, y) of two variables, x and y, and we want to know how f varies with only one of them, say x. To find out, we differentiate f with respect to x while treating the other variable y as a constant. The result is the partial derivative of f with respect to x, which is written fx f  df  x dx yconstant The rules for ordinary differentiation hold for partial differentiation as well. For instance, if f  cx2, df  2cx dx and so, if f  yx2, f  df   2yx x dx yconstant The partial derivative of f  yx2 with respect to the other variable, y, is f  df   x2 y dy xconstant Second order partial derivatives occur often in physics, as in the wave equation. To find 2fx2, we first calculate fx and then differentiate again, still keeping y constant: 2f  f x 2  x  x For f  yx2, 2f  2  (2yx)  2y x x 2f  2 Similarly  (x )  0 y2 y Solutions of the wave equation may be of many kinds, reflecting the variety of waves that can occur—a single traveling pulse, a train of waves of constant amplitude and wavelength, a train of superposed waves of the same amplitudes and wavelengths, a train of superposed waves of different amplitudes and wavelengths, bei48482_ch05.qxd 1/17/02 12:17 AM Page 165 Quantum Mechanics 165 y v A x y = A cos ω(t – x/v) Figure 5.1 Waves in the xy plane traveling in the x direction along a stretched string lying on the x axis. a standing wave in a string fastened at both ends, and so on. All solutions must be of the form x yF t   (5.4) where F is any function that can be differentiated. The solutions F(t  x) represent waves traveling in the x direction, and the solutions F(t  x) represent waves trav- eling in the x direction. Let us consider the wave equivalent of a “free particle,” which is a particle that is not under the influence of any forces and therefore pursues a straight path at constant speed. This wave is described by the general solution of Eq. (5.3) for undamped (that is, constant amplitude A), monochromatic (constant angular frequency ) harmonic waves in the x direction, namely y  Aei(tx) (5.5) In this formula y is a complex quantity, with both real and imaginary parts. Because ei  cos   i sin  Eq. (5.5) can be written in the form x x y  A cos  t     iA sin  t   (5.6) Only the real part of Eq. (5.6) [which is the same as Eq. (3.5)] has significance in the case of waves in a stretched string. There y represents the displacement of the string from its normal position (Fig. 5.1), and the imaginary part of Eq. (5.6) is discarded as irrelevant. Example 5.1 Verify that Eq. (5.5) is a solution of the wave equation. Solution The derivative of an exponential function eu is d u du (e )  eu dx dx The partial derivative of y with respect to x (which means t is treated as a constant) from Eq. (5.5) is therefore y i  y x  bei48482_ch05.qxd 1/17/02 12:17 AM Page 166 166 Chapter Five and the second partial derivative is 2y i22 2 2  2 y 2 y x   since i2  1. The partial derivative of y with respect to t (now holding x constant) is y  iy t and the second partial derivative is 2y  i22y  2y t2 Combining these results gives 2y 1 2y 2  x  2 t2 which is Eq. (5.3). Hence Eq. (5.5) is a solution of the wave equation. 5.3 SCHRÖDINGER’S EQUATION: TIME-DEPENDENT FORM A basic physical principle that cannot be derived from anything else In quantum mechanics the wave function  corresponds to the wave variable y of wave motion in general. However, , unlike y, is not itself a measurable quantity and may therefore be complex. For this reason we assume that  for a particle moving freely in the x direction is specified by   Aei(tx) (5.7) Replacing  in the above formula by 2 and  by  gives   Ae2i(tx) (5.8) This is convenient since we already know what  and  are in terms of the total energy E and momentum p of the particle being described by . Because h 2 E  h  2  and   p p we have Free particle   Ae(i )(Etpx) (5.9) Equation (5.9) describes the wave equivalent of an unrestricted particle of total energy E and momentum p moving in the x direction, just as Eq. (5.5) describes, for example, a harmonic displacement wave moving freely along a stretched string. The expression for the wave function  given by Eq. (5.9) is correct only for freely moving particles. However, we are most interested in situations where the motion of a particle is subject to various restrictions. An important concern, for example, is an electron bound to an atom by the electric field of its nucleus. What we must now do is obtain the fundamental differential equation for , which we can then solve for  in a specific situation. This equation, which is Schrödinger’s equation, can be arrived at in various ways, but it cannot be rigorously derived from existing physical principles: bei48482_ch05.qxd 1/17/02 12:17 AM Page 167 Quantum Mechanics 167 the equation represents something new. What will be done here is to show one route to the wave equation for  and then to discuss the significance of the result. We begin by differentiating Eq. (5.9) for  twice with respect to x, which gives 2 p2   2  x2 2 p2   2 (5.10) x2 Differentiating Eq. (5.9) once with respect to t gives  iE   t  E   (5.11) i t At speeds small compared with that of light, the total energy E of a particle is the sum of its kinetic energy p22m and its potential energy U, where U is in general a function of position x and time t: p2 E  U(x, t) (5.12) 2m The function U represents the influence of the rest of the universe on the particle. Of course, only a small part of the universe interacts with the particle to any extent; for Erwin Schrödinger (1887–1961) was thereby opening wide the door to the modern view of the atom born in Vienna to an Austrian father and which others had only pushed ajar. By June Schrödinger had a half-English mother and received his applied wave mechanics to the harmonic oscillator, the diatomic doctorate at the university there. After molecule, the hydrogen atom in an electric field, the absorption World War I, during which he served and emission of radiation, and the scattering of radiation by as an artillery officer, Schrödinger had atoms and molecules. He had also shown that his wave me- appointments at several German chanics was mathematically equivalent to the more abstract universities before becoming professor Heisenberg-Born-Jordan matrix mechanics. of physics in Zurich, Switzerland. Late The significance of Schrödinger’s work was at once realized. in November, 1925, Schrödinger gave a In 1927 he succeeded Planck at the University of Berlin but left talk on de Broglie’s notion that a moving particle has a wave Germany in 1933, the year he received the Nobel Prize, when character. A colleague remarked to him afterward that to deal the Nazis came to power. He was at Dublin’s Institute for Ad- properly with a wave, one needs a wave equation. Schrödinger vanced Study from 1939 until his return to Austria in 1956. In took this to heart, and a few weeks later he was “struggling with Dublin, Schrödinger became interested in biology, in particular a new atomic theory. If only I knew more mathematics! I am very the mechanism of heredity. He seems to have been the first to optimistic about this thing and expect that if I can only... solve make definite the idea of a genetic code and to identify genes it, it will be very beautiful.” (Schrödinger was not the only physicist as long molecules that carry the code in the form of variations to find the mathematics he needed difficult; the eminent mathe- in how their atoms are arranged. Schrödinger’s 1944 book What matician David Hilbert said at about this time, “Physics is much Is Life? was enormously influential, not only by what it said but too hard for physicists.”) also by introducing biologists to a new way of thinking—that The struggle was successful, and in January 1926 the first of of the physicist—about their subject. What Is Life? started James four papers on “Quantization as an Eigenvalue Problem” was Watson on his search for “the secret of the gene,” which he and completed. In this epochal paper Schrödinger introduced the Francis Crick (a physicist) discovered in 1953 to be the struc- equation that bears his name and solved it for the hydrogen atom, ture of the DNA molecule. bei48482_ch05.qxd 1/17/02 12:17 AM Page 168 168 Chapter Five instance, in the case of the electron in a hydrogen atom, only the electric field of the nucleus must be taken into account. Multiplying both sides of Eq. (5.12) by the wave function  gives p2 E   U (5.13) 2m Now we substitute for E and p2 from Eqs. (5.10) and (5.11) to obtain the time- dependent form of Schrödinger’s equation: Time-dependent Schrödinger  2 2 i   U (5.14) equation in one t 2m x2 dimension In three dimensions the time-dependent form of Schrödinger’s equation is  2 2  2 2 i  2  2   U (5.15) t 2m x y z2 where the particle’s potential energy U is some function of x, y, z, and t. Any restrictions that may be present on the particle’s motion will affect the potential- energy function U. Once U is known, Schrödinger’s equation may be solved for the wave function  of the particle, from which its probability density 2 may be de- termined for a specified x, y, z, t. Validity of Schrödinger’s Equation Schrödinger’s equation was obtained here using the wave function of a freely moving particle (potential energy U  constant). How can we be sure it applies to the general case of a particle subject to arbitrary forces that vary in space and time [U  U(x, y, z, t)]? Substituting Eqs. (5.10) and (5.11) into Eq. (5.13) is really a wild leap with no formal justification; this is true for all other ways in which Schrödinger’s equa- tion can be arrived at, including Schrödinger’s own approach. What we must do is postulate Schrödinger’s equation, solve it for a variety of phys- ical situations, and compare the results of the calculations with the results of experi- ments. If both sets of results agree, the postulate embodied in Schrödinger’s equation is valid. If they disagree, the postulate must be discarded and some other approach would then have to be explored. In other words, Schrödinger’s equation cannot be derived from other basic principles of physics; it is a basic principle in itself. What has happened is that Schrödinger’s equation has turned out to be remarkably accurate in predicting the results of experiments. To be sure, Eq. (5.15) can be used only for nonrelativistic problems, and a more elaborate formulation is needed when particle speeds near that of light are involved. But because it is in accord with experi- ence within its range of applicability, we must consider Schrödinger’s equation as a valid statement concerning certain aspects of the physical world. It is worth noting that Schrödinger’s equation does not increase the number of principles needed to describe the workings of the physical world. Newton’s second law bei48482_ch05.qxd 1/17/02 12:17 AM Page 169 Quantum Mechanics 169 of motion F  ma, the basic principle of classical mechanics, can be derived from Schrödinger’s equation provided the quantities it relates are understood to be averages rather than precise values. (Newton’s laws of motion were also not derived from any other principles. Like Schrödinger’s equation, these laws are considered valid in their range of applicability because of their agreement with experiment.) 5.4 LINEARITY AND SUPERPOSITION Wave functions add, not probabilities An important property of Schrödinger’s equation is that it is linear in the wave function . By this is meant that the equation has terms that contain  and its derivatives but no terms independent of  or that involve higher powers of  or its derivatives. As a result, a linear combination of solutions of Schrödinger’s equation for a given system is also itself a solution. If 1 and 2 are two solutions (that is, two wave functions that satisfy the equation), then   a11  a22 is also a solution, where a1 and a2 are constants (see Exercise 8). Thus the wave func- tions 1 and 2 obey the superposition principle that other waves do (see Sec. 2.1) and we conclude that interference effects can occur for wave functions just as they can for light, sound, water, and electromagnetic waves. In fact, the discussions of Secs. 3.4 and 3.7 assumed that de Broglie waves are subject to the superposition principle. Let us apply the superposition principle to the diffraction of an electron beam. Fig- ure 5.2a shows a pair of slits through which a parallel beam of monoenergetic elec- trons pass on their way to a viewing screen. If slit 1 only is open, the result is the intensity variation shown in Fig. 5.2b that corresponds to the probability density P1  12  *1 1 If slit 2 only is open, as in Fig. 5.2c, the corresponding probability density is P2  22  *2 2 We might suppose that opening both slits would give an electron intensity variation described by P1  P2, as in Fig. 5.2d. However, this is not the case because in quantum Electrons Screen Slit 2 Slit 1 2 2 2 2 2 Ψ1 Ψ2 Ψ1 + Ψ2 Ψ1 + Ψ2 (a) (b) (c) (d) (e) Figure 5.2 (a) Arrangement of double-slit experiment. (b) The electron intensity at the screen with only slit 1 open. (c) The electron intensity at the screen with only slit 2 open. (d) The sum of the intensities of (b) and (c). (e) The actual intensity at the screen with slits 1 and 2 both open. The wave functions 1 and 2 add to produce the intensity at the screen, not the probability densities 12 and 22. bei48482_ch05.qxd 1/17/02 12:17 AM Page 170 170 Chapter Five mechanics wave functions add, not probabilities. Instead the result with both slits open is as shown in Fig. 5.2e, the same pattern of alternating maxima and minima that oc- curs when a beam of monochromatic light passes through the double slit of Fig. 2.4. The diffraction pattern of Fig. 5.2e arises from the superposition  of the wave functions 1 and 2 of the electrons that have passed through slits 1 and 2:   1  2 The probability density at the screen is therefore P  2  1  22  (*1  *2 )(1  2)  *1 1  *2 2  *1 2  *2 1  P1  P2  *1 2  *2 1 The two terms at the right of this equation represent the difference between Fig. 5.2d and e and are responsible for the oscillations of the electron intensity at the screen. In Sec. 6.8 a similar calculation will be used to investigate why a hydrogen atom emits radiation when it undergoes a transition from one quantum state to another of lower energy. 5.5 EXPECTATION VALUES How to extract information from a wave function Once Schrödinger’s equation has been solved for a particle in a given physical situa- tion, the resulting wave function (x, y, z, t) contains all the information about the particle that is permitted by the uncertainty principle. Except for those variables that are quantized this information is in the form of probabilities and not specific numbers. As an example, let us calculate the expectation value x of the position of a particle confined to the x axis that is described by the wave function (x, t). This is the value of x we would obtain if we measured the positions of a great many particles described by the same wave function at some instant t and then averaged the results. To make the procedure clear, we first answer a slightly different question: What is the average position x of a number of identical particles distributed along the x axis in such a way that there are N1 particles at x1, N2 particles at x2, and so on? The average position in this case is the same as the center of mass of the distribution, and so N1x1  N2x2  N3 x3 ... Nixi x   (5.16) N1  N2  N3 ... Ni When we are dealing with a single particle, we must replace the number Ni of particles at xi by the probability Pi that the particle be found in an interval dx at xi. This probability is Pi  i2 dx (5.17) where i is the particle wave function evaluated at x  xi. Making this substitution and changing the summations to integrals, we see that the expectation value of the bei48482_ch05.qxd 1/17/02 12:17 AM Page 171 Quantum Mechanics 171 position of the single particle is  x dx   ___________ 2 x    dx  (5.18) 2  If  is a normalized wave function, the denominator of Eq. (5.18) equals the prob- ability that the particle exists somewhere between x   and x   and therefore has the value 1. In this case Expectation value for position x     x2 dx (5.19) Example 5.2 A particle limited to the x axis has the wave function   ax between x  0 and x  1;   0 elsewhere. (a) Find the probability that the particle can be found between x  0.45 and x  0.55. (b) Find the expectation value x of the particle’s position. Solution (a) The probability is  x2 x1 2 dx  a2  0.55 0.45 x2dx  a2 x3 3  0.55 0.45  0.0251a2 (b) The expectation value is x   x dx  a  x dx  a 0 1 2 2 1 0 3 2 x4 4  1 0  a2 4 The same procedure as that followed above can be used to obtain the expectation value G(x) of any quantity—for instance, potential energy U(x)—that is a function of the position x of a particle described by a wave function . The result is Expectation value G(x)     G(x)2 dx (5.20) The expectation value p for momentum cannot be calculated this way because, according to the uncertainty principles, no such function as p(x) can exist. If we specify x, so that x  0, we cannot specify a corresponding p since x p 2. The same problem occurs for the expectation value E for energy because E t 2 means that, if we specify t, the function E(t) is impossible. In Sec. 5.6 we will see how p and E can be determined. In classical physics no such limitation occurs, because the uncertainty principle can be neglected in the macroworld. When we apply the second law of motion to the motion of a body subject to various forces, we expect to get p(x, t) and E(x, t) from the solution as well as x(t). Solving a problem in classical mechanics gives us the en- tire future course of the body’s motion. In quantum physics, on the other hand, all we get directly by applying Schrödinger’s equation to the motion of a particle is the wave function , and the future course of the particle’s motion—like its initial state—is a matter of probabilities instead of certainties. bei48482_ch05.qxd 1/17/02 12:17 AM Page 172 172 Chapter Five 5.6 OPERATORS Another way to find expectation values A hint as to the proper way to evaluate p and E comes from differentiating the free- particle wave function   Ae(i )(Etpx) with respect to x and to t. We find that  i  p x  i   E t which can be written in the suggestive forms  p   (5.21) i x  E  i  (5.22) t Evidently the dynamical quantity p in some sense corresponds to the differential operator ( i) x and the dynamical quantity E similarly corresponds to the differ- ential operator i t. An operator tells us what operation to carry out on the quantity that follows it. Thus the operator i t instructs us to take the partial derivative of what comes after it with respect to t and multiply the result by i. Equation (5.22) was on the postmark used to cancel the Austrian postage stamp issued to commemorate the 100th anniversary of Schrödinger’s birth. It is customary to denote operators by using a caret, so that p̂ is the operator that corresponds to momentum p and Ê is the operator that corresponds to total energy E. From Eqs. (5.21) and (5.22) these operators are Momentum  p̂  (5.23) operator i x Total-energy  operator Ê  i (5.24) t Though we have only shown that the correspondences expressed in Eqs. (5.23) and (5.24) hold for free particles, they are entirely general results whose validity is the same as that of Schrödinger’s equation. To support this statement, we can re- place the equation E  KE  U for the total energy of a particle with the operator equation Ê  KˆE  Û (5.25) The operator Û is just U (). The kinetic energy KE is given in terms of momen- tum p by p2 KE  2m bei48482_ch05.qxd 1/17/02 12:17 AM Page 173 Quantum Mechanics 173 and so we have p̂2 1  2 2  2 Kinetic-energy KˆE    (5.26) operator 2m 2m i x 2m x2 Equation (5.25) therefore reads  2 2 i  U (5.27) t 2m x2 Now we multiply the identity    by Eq. (5.27) and obtain  2 2 i   U t 2m x2 which is Schrödinger’s equation. Postulating Eqs. (5.23) and (5.24) is equivalent to postulating Schrödinger’s equation. Operators and Expectation Values Because p and E can be replaced by their corresponding operators in an equation, we can use these operators to obtain expectation values for p and E. Thus the expectation value for p is p    *p̂ dx     *   i x  dx  i    *  x dx (5.28) and the expectation value for E is E   *Ê dx   * i    t  dx  i  * t dx  (5.29) Both Eqs. (5.28) and (5.29) can be evaluated for any acceptable wave function  (x, t). Let us see why expectation values involving operators have to be expressed in the form p    *p̂ dx The other alternatives are   p̂* dx  i     x (*) dx  i *    0 since * and  must be 0 at x  , and    * p̂ dx  i   *  x dx which makes no sense. In the case of algebraic quantities such as x and V(x), the order of factors in the integrand is unimportant, but when differential operators are involved, the correct order of factors must be observed. bei48482_ch05.qxd 1/17/02 12:17 AM Page 174 174 Chapter Five Every observable quantity G characteristic of a physical system may be represented by a suitable quantum-mechanical operator Ĝ. To obtain this operator, we express G in terms of x and p and then replace p by ( i) x. If the wave function  of the system is known, the expectation value of G(x, p) is Expectation value of an operator G(x, p)     *Ĝ dx (5.30) In this way all the information about a system that is permitted by the uncertainty principle can be obtained from its wave function . 5.7 SCHRÖDINGER’S EQUATION: STEADY-STATE FORM Eigenvalues and eigenfunctions In a great many situations the potential energy of a particle does not depend on time explicitly; the forces that act on it, and hence U, vary with the position of the particle only. When this is true, Schrödinger’s equation may be simplified by removing all reference to t. We begin by noting that the one-dimensional wave function  of an unrestricted particle may be written   Ae(i )(Etpx)  Ae(iE )te(ip )x  e(iE )t (5.31) Evidently  is the product of a time-dependent function e(iE )t and a position- dependent function. As it happens, the time variations of all wave functions of particles acted on by forces independent of time have the same form as that of an unrestricted particle. Substituting the  of Eq. (5.31) into the time-dependent form of Schrödinger’s equation, we find that 2 2 E e(iE )t  e(iE )t  U e(iE )t 2m x2 Dividing through by the common exponential factor gives Steady-state Schrödinger equation 2 2m (5.32) 2  2 (E  U)  0 in one dimension x Equation (5.32) is the steady-state form of Schrödinger’s equation. In three dimen- sions it is Steady-state Schrödinger 2 2 2 2m 2  2   2 (E  U)  0 (5.33) equation in three x y z2 dimensions An important property of Schrödinger’s steady-state equation is that, if it has one or more solutions for a given system, each of these wave functions corresponds to a specific value of the energy E. Thus energy quantization appears in wave mechanics as a natural element of the theory, and energy quantization in the physical world is re- vealed as a universal phenomenon characteristic of all stable systems. bei48482_ch05.qxd 1/17/02 12:17 AM Page 175 Quantum Mechanics 175 A familiar and quite close analogy to the manner in which energy quantization occurs in solutions of Schrödinger’s equation is with standing waves in a stretched string of length L that is fixed at both ends. Here, instead of a single wave propagating indefi- nitely in one direction, waves are traveling in both the x and x directions simul- λ = 2L taneously. These waves are subject to the condition (called a boundary condition) that the displacement y always be zero at both ends of the string. An acceptable function λ=L y(x, t) for the displacement must, with its derivatives (except at the ends), be as well- behaved as and its derivatives—that is, be continuous, finite, and single-valued. In λ = 2L this case y must be real, not complex, as it represents a directly measurable quantity. 3 The only solutions of the wave equation, Eq. (5.3), that are in accord with these various limitations are those in which the wavelengths are given by λ = 1L 2 2L L n  n  0, 1, 2, 3,... n1 λ = 2L n = 0, 1, 2, 3,... n+1 as shown in Fig. 5.3. It is the combination of the wave equation and the restrictions Figure 5.3 Standing waves in a placed on the nature of its solution that leads us to conclude that y(x, t) can exist only stretched string fastened at both ends. for certain wavelengths n. Eigenvalues and Eigenfunctions The values of energy En for which Schrödinger’s steady-state equation can be solved are called eigenvalues and the corresponding wave functions n are called eigen- functions. (These terms come from the German Eigenwert, meaning “proper or char- acteristic value,” and Eigenfunktion, “proper or characteristic function.”) The discrete energy levels of the hydrogen atom me4 n 1 En   n  1, 2, 3,... 322 20 2 2 are an example of a set of eigenvalues. We shall see in Chap. 6 why these particular values of E are the only ones that yield acceptable wave functions for the electron in the hydrogen atom. An important example of a dynamical variable other than total energy that is found to be quantized in stable systems is angular momentum L. In the case of the hydro- gen atom, we shall find that the eigenvalues of the magnitude of the total angular momentum are specified by ) L  l(l  1 l  0, 1, 2,... , (n  1) Of course, a dynamical variable G may not be quantized. In this case measurements of G made on a number of identical systems will not yield a unique result but instead a spread of values whose average is the expectation value G    G 2 dx In the hydrogen atom, the electron’s position is not quantized, for instance, so that we must think of the electron as being present in the vicinity of the nucleus with a cer- tain probability  2 per unit volume but with no predictable position or even orbit in the classical sense. This probabilistic statement does not conflict with the fact that bei48482_ch05.qxd 1/17/02 12:17 AM Page 176 176 Chapter Five experiments performed on hydrogen atoms always show that each one contains a whole electron, not 27 percent of an electron in a certain region and 73 percent elsewhere. The probability is one of finding the electron, and although this probability is smeared out in space, the electron itself is not. Operators and Eigenvalues The condition that a certain dynamical variable G be restricted to the discrete values Gn—in other words, that G be quantized—is that the wave functions n of the system be such that Eigenvalue equation Ĝ n  Gn n (5.34) where Ĝ is the operator that corresponds to G and each Gn is a real number. When Eq. (5.34) holds for the wave functions of a system, it is a fundamental postulate of quantum mechanics that any measurement of G can only yield one of the values Gn. If measurements of G are made on a number of identical systems all in states described by the particular eigenfunction k , each measurement will yield the single value Gk. Example 5.3 An eigenfunction of the operator d2dx2 is  e2x. Find the corresponding eigenvalue. Solution Here Ĝ  d2dx2, so d2 2x d d 2x d Ĝ  dx2 (e )  dx dx2 (e )   dx (2e2x)  4e2x But e2x  , so Ĝ  4 From Eq. (5.34) we see that the eigenvalue G here is just G  4. In view of Eqs. (5.25) and (5.26) the total-energy operator Ê of Eq. (5.24) can also be written as Hamiltonian 2 2 Ĥ   U (5.35) operator 2m x2 and is called the Hamiltonian operator because it is reminiscent of the Hamiltonian function in advanced classical mechanics, which is an expression for the total energy of a system in terms of coordinates and momenta only. Evidently the steady-state Schrödinger equation can be written simply as Schrödinger’s equation Ĥ n  En n (5.36) bei48482_ch05.qxd 1/17/02 12:17 AM Page 177 Quantum Mechanics 177 Table 5.1 Operators Associated with Various Observable Quantities Quantity Operator Position, x x  Linear momentum, p i x Potential energy, U(x) U(x) p2 2 2 Kinetic energy, KE   2m 2m x2  Total energy, E i t 2 2 Total energy (Hamiltonian form), H   U(x) 2m x2 so we can say that the various En are the eigenvalues of the Hamiltonian operator Ĥ. This kind of association between eigenvalues and quantum-mechanical operators is quite general. Table 5.1 lists the operators that correspond to various observable quantities. 5.8 PARTICLE IN A BOX How boundary conditions and normalization determine wave functions To solve Schrödinger’s equation, even in its simpler steady-state form, usually requires elaborate mathematical techniques. For this reason the study of quantum mechanics has traditionally been reserved for advanced students who have the required profi- ciency in mathematics. However, since quantum mechanics is the theoretical structure whose results are closest to experimental reality, we must explore its methods and ap- plications to understand modern physics. As we shall see, even a modest mathemati- cal background is enough for us to follow the trains of thought that have led quantum mechanics to its greatest achievements. The simplest quantum-mechanical problem is that of a particle trapped in a box with infinitely hard walls. In Sec. 3.6 we saw how a quite simple argument yields the energy levels of the system. Let us now tackle the same problem in a more formal way, which will give us the wave function n that corresponds to each energy level. We may specify the particle’s motion by saying that it is restricted to traveling along ∞ the x axis between x  0 and x  L by infintely hard walls. A particle does not lose U energy when it collides with such walls, so that its total energy stays constant. From a formal point of view the potential energy U of the particle is infinite on both sides of the box, while U is a constant—say 0 for convenience—on the inside (Fig. 5.4). Because the particle cannot have an infinite amount of energy, it cannot exist outside the box, and so its wave function is 0 for x  0 and x L. Our task is to find what is x within the box, namely, between x  0 and x  L. 0 L Within the box Schrödinger’s equation becomes Figure 5.4 A square potential well 2 with infinitely high barriers at d 2m  E 0 (5.37) each end corresponds to a box dx2 2 with infinitely hard walls. bei48482_ch05.qxd 1/17/02 12:17 AM Page 178 178 Chapter Five since U  0 there. (The total derivative d2 dx2 is the same as the partial derivative 2 x2 because is a function only of x in this problem.) Equation (5.37) has the solution  2mE 2mE   A sin x  B cos x (5.38) which we can verify by substitution back into Eq. (5.37). A and B are constants to be evaluated. This solution is subject to the boundary conditions that  0 for x  0 and for x  L. Since cos 0  1, the second term cannot describe the particle because it does not vanish at x  0. Hence we conclude that B  0. Since sin 0  0, the sine term always yields  0 at x  0, as required, but will be 0 at x  L only when 2mE  L  n n  1, 2, 3,... (5.39) This result comes about because the sines of the angles , 2, 3,... are all 0. From Eq. (5.39) it is clear that the energy of the particle can have only certain val- ues, which are the eigenvalues mentioned in the previous section. These eigenvalues, constituting the energy levels of the system, are found by solving Eq. (5.39) for En, which gives n22 2 Particle in a box En  n  1, 2, 3,... (5.40) 2mL2 Equation (5.40) is the same as Eq. (3.18) and has the same interpretation [see the discussion that follows Eq. (3.18) in Sec. 3.6]. Wave Functions The wave functions of a particle in a box whose energies are En are, from Eq. (5.38) with B  0, n 2mE n  A sin x (5.41) Substituting Eq. (5.40) for En gives nx n  A sin (5.42) L for the eigenfunctions corresponding to the energy eigenvalues En. It is easy to verify that these eigenfunctions meet all the requirements discussed in Sec. 5.1: for each quantum number n, n is a finite, single-valued function of x, and n and  nx are continuous (except at the ends of the box). Furthermore, the integral bei48482_ch05.qxd 1/17/02 12:17 AM Page 179 Quantum Mechanics 179 of  n2 over all space is finite, as we can see by integrating  n2 dx from x  0 to x  L (since the particle is confined within these limits). With the help of the trigonometric identity sin2   12 (1  cos 2) we find that     n2 dx   L 0 n 2 dx  A2  L 0 sin2  nx L dx  A2 2  dx   cos  2nx L 0 L 0 L dx A2 L   L 2nx L  2 x 2n sin L 0  A2 2 (5.43) To normalize we must assign a value to A such that  n2 dx is equal to the prob- 3 ability P dx of finding the particle between x and x  dx, rather than merely propor- tional to P dx. If  n2 dx is to equal P dx, then it must be true that    n2 dx  1 (5.44) 2 Comparing Eqs. (5.43) and (5.44), we see that the wave functions of a particle in a box are normalized if L 2 A (5.45) 1 x=0 x=L The normalized wave functions of the particle are therefore L sin 2 nx Particle in a box n  n  1, 2, 3,... (5.46) 2 L | 3| The normalized wave functions 1, 2, and 3 together with the probability densities  12,  22, and  32 are plotted in Fig. 5.5. Although n may be negative as well as positive,  n2 is never negative and, since n is normalized, its value at a given x is | 2 equal to the probability density of finding the particle there. In every case  n2  0 at 2| x  0 and x  L, the boundaries of the box. At a particular place in the box the probability of the particle being present may be very different for different quantum numbers. For instance,  12 has its maximum value of 2L in the middle of the box, while  22  0 there. A particle in the lowest energy level of n  1 is most likely to be in the middle of the box, while a particle in | 1| 2 the next higher state of n  2 is never there! Classical physics, of course, suggests the same probability for the particle being anywhere in the box. The wave functions shown in Fig. 5.5 resemble the possible vibrations of a string x=0 x=L fixed at both ends, such as those of the stretched string of Fig. 5.2. This follows from the fact that waves in a stretched string and the wave representing a moving particle Figure 5.5 Wave functions and are described by equations of the same form, so that when identical restrictions are probability densities of a particle placed upon each kind of wave, the formal results are identical. confined to a box with rigid walls. bei48482_ch05.qxd 2/6/02 6:51 PM Page 180 180 Chapter Five Example 5.4 Find the probability that a particle trapped in a box L wide can be found between 0.45L and 0.55L for the ground and first excited states. Solution This part of the box is one-tenth of the box’s width and is centered on the middle of the box (Fig. 5.6). Classically we would expect the particle to be in this region 10 percent of the time. Quantum mechanics gives quite different predictions that depend on the quantum number of the particle’s state. From Eqs. (5.2) and (5.46) the probability of finding the particle between x1 and x2 when it is in the nth state is Px1, x2   x2 x1 2 n2 dx   L  x2 x1 nx sin2  dx L x2   x 1 2nx     sin  L 2n L x1 Here x1  0.45L and x2  0.55L. For the ground state, which corresponds to n  1, we have Px1, x2  0.198  19.8 percent This is about twice the classical probability. For the first excited state, which corresponds to n  2, we have Px1, x2  0.0065  0.65 percent This low figure is consistent with the probability density of n2  0 at x  0.5L. | 2|2 | 1|2 x=0 x1 x2 x=L Figure 5.6 The probability Px1, x2 of finding a particle in the box of Fig. 5.5 between x1  0.45L and x2  0.55L is equal to the area under the 2 curves between these limits. bei48482_ch05.qxd 1/17/02 12:17 AM Page 181 Quantum Mechanics 181 Example 5.5 Find the expectation value x of the position of a particle trapped in a box L wide. Solution From Eqs. (5.19) and (5.46) we have x     x 2 dx  2 L  L 0 x sin2 nx L dx x2  cos(2nxL) L 2 x sin(2nxL)    L 4 4nL 8(nL)2 0 Since sin n  0, cos 2n  1, and cos 0  1, for all the values of n the expectation value of x is L2 4 2 L x   L 2 This result means that the average position of the particle is the middle of the box in all quan- tum states. There is no conflict with the fact that  2  0 at L2 in the n  2, 4, 6,... states because x is an average, not a probability, and it reflects the symmetry of  2 about the middle of the box. Momentum Finding the momentum of a particle trapped in a one-dimensional box is not as straight- forward as finding x. Here L sin 2 nx * n  L L L d 2 n nx  cos dx L and so, from Eq. (5.30),