6 Chapter 1 The Schrodinger Equation using the wave or particle concept of classical physics.The concepts of classical phys- ics have been developed from experience in the macroscopic world and do not properly describe the microscopic world.Evolution has shaped the human brain to allow it to un- derstand and deal effectively with macroscopic phenomena.The human nervous system was not developed to deal with phenomena at the atomic and molecular level,so it is not surprising if we cannot fully understand such phenomena. Although both photons and electrons show an apparer t duality,they are not the sam kinds of entities.Photons travel at speedc and a nonzero res 1V1s0 1.3 The Uncertainty Principle Let us consider what effect the wave-particle duality has on attempts to measure simulta neously the x coordinate and thex component of linear momentum of a microscopic par- ticle.We start with a beam of particles with momentum p.traveling in the y direction.and we let the beam fall on a narrow slit.Behind this slit is a photographic plate.See Fig.1.1. Particles that pass through the slit o whave an uncertainty w in their x coor cop particles hay wave properties,th ey a e diffracted by the slit pro a light be n)a enh e plate n sho s that when the icles acted by the slit.their direction of mo as so that part of their momentum was transferred to the direction.The component of momentum p,equals the projection of the momentum vector p in thex di- rection A narticle deflected uward by an angle has n=p sin o A narticle deflected downward by a has=-sin a Since most of the particles undergo deflections in the range-a to a,where a is the angle to the first minimum in the diffraction pattern,we shall take one-half the spread of momentum values in the central diffraction peak as a measure of the uncertainty Ap,in the x component of momentum:Ap,=p sin a. Hence at the slit.where the measurement is made. △r△p,-pw sina (1.6) EeEgDfactonot Photographic plate
6 Chapter 1 | The Schrödinger Equation using the wave or particle concept of classical physics. The concepts of classical physics have been developed from experience in the macroscopic world and do not properly describe the microscopic world. Evolution has shaped the human brain to allow it to understand and deal effectively with macroscopic phenomena. The human nervous system was not developed to deal with phenomena at the atomic and molecular level, so it is not surprising if we cannot fully understand such phenomena. Although both photons and electrons show an apparent duality, they are not the same kinds of entities. Photons travel at speed c in vacuum and have zero rest mass; electrons always have v 6 c and a nonzero rest mass. Photons must always be treated relativistically, but electrons whose speed is much less than c can be treated nonrelativistically. 1.3 The Uncertainty Principle Let us consider what effect the wave–particle duality has on attempts to measure simultaneously the x coordinate and the x component of linear momentum of a microscopic particle. We start with a beam of particles with momentum p, traveling in the y direction, and we let the beam fall on a narrow slit. Behind this slit is a photographic plate. See Fig. 1.1. Particles that pass through the slit of width w have an uncertainty w in their x coordinate at the time of going through the slit. Calling this spread in x values x, we have x = w. Since microscopic particles have wave properties, they are diffracted by the slit producing (as would a light beam) a diffraction pattern on the plate. The height of the graph in Fig. 1.1 is a measure of the number of particles reaching a given point. The diffraction pattern shows that when the particles were diffracted by the slit, their direction of motion was changed so that part of their momentum was transferred to the x direction. The x component of momentum px equals the projection of the momentum vector p in the x direction. A particle deflected upward by an angle a has px = p sin a. A particle deflected downward by a has px = -p sin a. Since most of the particles undergo deflections in the range -a to a, where a is the angle to the first minimum in the diffraction pattern, we shall take one-half the spread of momentum values in the central diffraction peak as a measure of the uncertainty px in the x component of momentum: px = p sin a. Hence at the slit, where the measurement is made, x px = pw sin a (1.6) p w p sin a A x y E Photographic plate a a a p Figure 1.1 Diffraction of electrons by a slit
1.4 The Time-Dependent Schrodinger Equation7 rst a The first diffr ction minimum occurs is readily calculated The es ta led by t eof the ppe and pa of th riginating from the top of the slit are then exactly with wa ves nating fro n the center of the slit.and the cancel each other.Waves originati na f a point in the slit at a distanced below the slit midpoint cancel with waves orignating at a distance d below the ton of the slit drawing acin fig 12 so that ad=CD we have the difference in path length as BC.The distance from the slit to the screen is large compared with the slit width.Hence AD and BD are nearly parallel.This makes the angle ACB essentially a right angle,and so angle BAC =a.The path difference BC is then w sin a.Setting BC equal to入，we have wsin a=入，and Eq.(l.6)be comes Ax Ap,pA.The wavelength A is given by the de Broglie relation =h/p.so Ar Ap,=h.Since the uncertainties have not been precisely defined,the equality sign is not really justified.Instead we write △x△p=h (1.7) e product of the uncertainties inand p is of the order of magnitude of Although we have demor nstrated (17)for is general.No matter what attempts are made.the rticle duality of mic 、a limit on our ability to mea simultan osition and m enofhpaasTeenoeeeehlusyhcDetesm ure is our determination of momentum.(In Fig.1.1,sin a=A/w,so narrowing the slit in- creases the spread of the diffraction pattern.)This limitation is the uncertainty principle. discovered in 1927 by Werner Heisenberg. Because of the wave-particle duality,the act of measurement introduces an uncon- trollable disturbance in the system being measured.We started with particles having a precise value of p.(zero).By imposing the slit,we measured thex coordinate of the par sureme t introduce I an uncertainty into the p,values 1.4 The Time-Dependent Schrodinger Equation between c cal and qu one-dimensional system will be e discussed
1.4 The Time-Dependent Schrödinger Equation | 7 The angle a at which the first diffraction minimum occurs is readily calculated. The condition for the first minimum is that the difference in the distances traveled by particles passing through the slit at its upper edge and particles passing through the center of the slit should be equal to 1 2 l, where l is the wavelength of the associated wave. Waves originating from the top of the slit are then exactly out of phase with waves originating from the center of the slit, and they cancel each other. Waves originating from a point in the slit at a distance d below the slit midpoint cancel with waves originating at a distance d below the top of the slit. Drawing AC in Fig. 1.2 so that AD = CD, we have the difference in path length as BC. The distance from the slit to the screen is large compared with the slit width. Hence AD and BD are nearly parallel. This makes the angle ACB essentially a right angle, and so angle BAC = a. The path difference BC is then 1 2 w sin a. Setting BC equal to 1 2 l, we have w sin a = l, and Eq. (1.6) becomes x px = pl. The wavelength l is given by the de Broglie relation l = h>p, so x px = h. Since the uncertainties have not been precisely defined, the equality sign is not really justified. Instead we write x px h (1.7) indicating that the product of the uncertainties in x and px is of the order of magnitude of Planck’s constant. Although we have demonstrated (1.7) for only one experimental setup, its validity is general. No matter what attempts are made, the wave–particle duality of microscopic “particles” imposes a limit on our ability to measure simultaneously the position and momentum of such particles. The more precisely we determine the position, the less accurate is our determination of momentum. (In Fig. 1.1, sin a = l>w, so narrowing the slit increases the spread of the diffraction pattern.) This limitation is the uncertainty principle, discovered in 1927 by Werner Heisenberg. Because of the wave–particle duality, the act of measurement introduces an uncontrollable disturbance in the system being measured. We started with particles having a precise value of px (zero). By imposing the slit, we measured the x coordinate of the particles to an accuracy w, but this measurement introduced an uncertainty into the px values of the particles. The measurement changed the state of the system. 1.4 The Time-Dependent Schrödinger Equation Classical mechanics applies only to macroscopic particles. For microscopic “particles” we require a new form of mechanics, called quantum mechanics. We now consider some of the contrasts between classical and quantum mechanics. For simplicity a one-particle, one-dimensional system will be discussed. Figure 1.2 Calculation of first diffraction minimum. A w B E C D a a 1 2
8 Chapter 1 The Schrodinger Equation In classical mechanics the motion of a particle is governed by Newton's second law: F=ma =m d 1.8) and t is the time:a is the ac (d/dt)(dx/dt)=d2x/dr2,where v is the velocity. Equation (1.8)contains the second derivative of the coordinatex with respect to time.To solve it,we must carry out two integrations.This introduces two arbitrary constants c and c2 into the solution,and x=g(t.cI.c2) 1.9 where g is some function of time.We now ask:What information must we possess at a given time to to be able to predict the future motion of the particle?If we know that at to the particle is at point xo.we have xo=g(to.c1.c2) (1.10) If we also know that at time to the particle has velocity then we have the additional relation d 1.1 = We may then use (1.10)and (1.11)to solve for c and c2 in terms of xo and vo Knowing c and c2.we can use Eq.(1.9)to predict the exact future motion of the particle As an example of Eqs.(1.8)to (1.11).consider the vertical motion of a particle in the earth's gravitational field.Let theaxis point upwar The force on the particle is .where g the gr md-x/d on give itrary cons ant c d/dcy which ed if we kn had )2+v0(t wing xo and vo The classical-mechanical potential energy Vof a particle moving in one dimension is defined to satisfy aV(x.t)=-F(x.t) 1.12) ax For example,for a particle moving in the earth's gravitational field,av/ax =-F=mg and integration gives V=mgx +c,where c is an arbitrary constant.We are free to set the zero level of potential energy wherever we please.Choosing c =0.we have V=mgx as the potential-energy function The word state in classical mechanics means a specification of the position and veloc- ity of each particle of the system at some instant of time,plus specification of the forces
8 Chapter 1 | The Schrödinger Equation In classical mechanics the motion of a particle is governed by Newton’s second law: F = ma = m d 2 x dt 2 (1.8) where F is the force acting on the particle, m is its mass, and t is the time; a is the acceleration, given by a = dv>dt = 1d>dt21dx>dt2 = d 2 x>dt 2 , where v is the velocity. Equation (1.8) contains the second derivative of the coordinate x with respect to time. To solve it, we must carry out two integrations. This introduces two arbitrary constants c1 and c2 into the solution, and x = g1t, c1, c22 (1.9) where g is some function of time. We now ask: What information must we possess at a given time t0 to be able to predict the future motion of the particle? If we know that at t0 the particle is at point x0, we have x0 = g1t0, c1, c22 (1.10) Since we have two constants to determine, more information is needed. Differentiating (1.9), we have dx dt = v = d dt g1t, c1, c22 If we also know that at time t0 the particle has velocity v0, then we have the additional relation v0 = d dt g1t, c1, c22 ` t=t0 (1.11) We may then use (1.10) and (1.11) to solve for c1 and c2 in terms of x0 and v0. Knowing c1 and c2, we can use Eq. (1.9) to predict the exact future motion of the particle. As an example of Eqs. (1.8) to (1.11), consider the vertical motion of a particle in the earth’s gravitational field. Let the x axis point upward. The force on the particle is downward and is F = -mg, where g is the gravitational acceleration constant. Newton’s second law (1.8) is -mg = md 2 x>dt 2 , so d 2 x>dt 2 = -g. A single integration gives dx>dt = -gt + c1. The arbitrary constant c1 can be found if we know that at time t0 the particle had velocity v0. Since v = dx>dt, we have v0 = -gt0 + c1 and c1 = v0 + gt0. Therefore, dx>dt = -gt + gt0 + v0. Integrating a second time, we introduce another arbitrary constant c2, which can be evaluated if we know that at time t0 the particle had position x0. We find (Prob. 1.7) x = x0 - 1 2 g1t - t022 + v01t - t02. Knowing x0 and v0 at time t0, we can predict the future position of the particle. The classical-mechanical potential energy V of a particle moving in one dimension is defined to satisfy 0V1x, t2 0x = -F1x, t2 (1.12) For example, for a particle moving in the earth’s gravitational field, 0V>0x = -F = mg and integration gives V = mgx + c, where c is an arbitrary constant. We are free to set the zero level of potential energy wherever we please. Choosing c = 0, we have V = mgx as the potential-energy function. The word state in classical mechanics means a specification of the position and velocity of each particle of the system at some instant of time, plus specification of the forces
1.4 The Time-Dependent Schrodinger Equation9 acting on the particles.According to Newton's second law.given the state of a systemat ons are exactly ,as shown by laws in ex ary motions The nd 17A01Q hat the ton's la universe at some instant,the future tion of eve ything in the u rse was co determined.A super-being able to know the state of the universe at any instant could.i principle,calculate all future motions. Although al-mechanical systems eriodically yary he initial values c ntanym Thus.be which one can measure the initial state is limited prediction of the long-term behavion of a chaotic classical- -mechanical system is,in practice,impossible,even though the h man,1993: J.J.Lissauer.Rev.Mod.Phys.71.835 (1999)1. Giv wledge of the P nt state of a classical-m echanical sy a er th inty princ the veryk P op par s fo tent i with ething less than com rediction of the exact future Our app roach to quantum mechanics will be to postulate the basic principles and then use these postulates to deduce experimentally testable consequences such as the energy levels of atoms.To describe the state of a system in quantum mechanics,we postulate the existence of a function w of the particles'coordinates called the state function or wave function (often written as wavefunction).Since the state will,in general,change with time,Y is also a function of time.For a one-particle,one-dimensional system,we have (x.t).The wave function contains all possible informati on about a system o instead of speaking of“"the state described by the wave functionΨ，”we simply say the state Newton's second law tells us how to find the futur mechanic ge or its pr t state state system I nges with time. tulated to be a one-particle. is pos Ψ(x,) 2a2Ψ(xt) (1.13 2m ax2 +V(x,t)Ψ(xt) where the constanth(h-bar)is defined as 方 1.14) 2
1.4 The Time-Dependent Schrödinger Equation | 9 acting on the particles. According to Newton’s second law, given the state of a system at any time, its future state and future motions are exactly determined, as shown by Eqs. (1.9)–(1.11). The impressive success of Newton’s laws in explaining planetary motions led many philosophers to use Newton’s laws as an argument for philosophical determinism. The mathematician and astronomer Laplace (1749–1827) assumed that the universe consisted of nothing but particles that obeyed Newton’s laws. Therefore, given the state of the universe at some instant, the future motion of everything in the universe was completely determined. A super-being able to know the state of the universe at any instant could, in principle, calculate all future motions. Although classical mechanics is deterministic, many classical-mechanical systems (for example, a pendulum oscillating under the influence of gravity, friction, and a periodically varying driving force) show chaotic behavior for certain ranges of the systems’ parameters. In a chaotic system, the motion is extraordinarily sensitive to the initial values of the particles’ positions and velocities and to the forces acting, and two initial states that differ by an experimentally undetectable amount will eventually lead to very different future behavior of the system. Thus, because the accuracy with which one can measure the initial state is limited, prediction of the long-term behavior of a chaotic classical-mechanical system is, in practice, impossible, even though the system obeys deterministic equations. Computer calculations of solar-system planetary orbits over tens of millions of years indicate that the motions of the planets are chaotic [I. Peterson, Newton’s Clock: Chaos in the Solar System, Freeman, 1993; J. J. Lissauer, Rev. Mod. Phys., 71, 835 (1999)]. Given exact knowledge of the present state of a classical-mechanical system, we can predict its future state. However, the Heisenberg uncertainty principle shows that we cannot determine simultaneously the exact position and velocity of a microscopic particle, so the very knowledge required by classical mechanics for predicting the future motions of a system cannot be obtained. We must be content in quantum mechanics with something less than complete prediction of the exact future motion. Our approach to quantum mechanics will be to postulate the basic principles and then use these postulates to deduce experimentally testable consequences such as the energy levels of atoms. To describe the state of a system in quantum mechanics, we postulate the existence of a function of the particles’ coordinates called the state function or wave function (often written as wavefunction). Since the state will, in general, change with time, is also a function of time. For a one-particle, one-dimensional system, we have = 1x, t2. The wave function contains all possible information about a system, so instead of speaking of “the state described by the wave function ,” we simply say “the state .” Newton’s second law tells us how to find the future state of a classicalmechanical system from knowledge of its present state. To find the future state of a quantum-mechanical system from knowledge of its present state, we want an equation that tells us how the wave function changes with time. For a one-particle, one-dimensional system, this equation is postulated to be - U i 0 1x, t2 0t = - U2 2m 02 1x, t2 0x2 + V1x, t2 1x, t2 (1.13) where the constant U (h-bar) is defined as U K h 2p (1.14)
10 Chapter 1 The Schrodinger Equation The concept of the wave function and the equation governing its change with time were discovered in 1926 by the Austrian physicist Erwin Schrodinger(1887-1961).In this equation,known as the time-dependent Schrodinger equation (or the Schrodinger wave equation),i=V-1.m is the mass of the particle,and V(x,t)is the potential- energy function of the system.(Many of the historically important papers in quantum mechanics are available at dieumsnh.qfb.umich.mx/archivoshistoricosmq.) The tir -depender of the wav any e.ifw he e to. wave fur cti tem it de ontains all the inf n we can possibly know about the sys do ticle?We er to thie stion was provided by Max Born shortly after Schrodinger discovered the Schrodinger 1.15) gives the probabiliry at timet of finding the particle in the region of the x axis ly- ing between x and x dr.In (1.15)the bars der note the absolute value and dx is an infinit lengt the x axis Ψx,) s the proba ilit den n16 cle at various places on the xa (Pro ty is re e.supp ed h some particular to th an s po 40- where an se t r are the nore likely to be found than other alus. um at the origin in this case To relate2 to experimental measurements,we would take many identical non- interacting systems,each of which was in the same state V.Then the particle's position in each system is measured.If we had n systems and made n measurements,and if dn denotes the number of measurements for which we found the particle between x and x+dt.then dn,/n is the probability for finding the particle between x and x+dx.Thus =2d hat taki ne cus m that was in state v and ro easuring the rticle' sition Thi procedure is wrong because the process of measurement ges the state of a system.We saw an example of this in the discussion of the uncertainty principle (Section 1 3) Quantum mechanics is statistical in nature.Knowing the state.we cannot predict the result of a position measurement with certainty:we can only predict the probabilities of various possible results.The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture. Quantum mechanics does not say that an electron is distributed over a large region of her.it is the probability patterns(wave functions)used
10 Chapter 1 | The Schrödinger Equation The concept of the wave function and the equation governing its change with time were discovered in 1926 by the Austrian physicist Erwin Schrödinger (1887–1961). In this equation, known as the time-dependent Schrödinger equation (or the Schrödinger wave equation), i = 2-1, m is the mass of the particle, and V1x, t2 is the potentialenergy function of the system. (Many of the historically important papers in quantum mechanics are available at dieumsnh.qfb.umich.mx/archivoshistoricosmq.) The time-dependent Schrödinger equation contains the first derivative of the wave function with respect to time and allows us to calculate the future wave function (state) at any time, if we know the wave function at time t0. The wave function contains all the information we can possibly know about the system it describes. What information does give us about the result of a measurement of the x coordinate of the particle? We cannot expect to involve the definite specification of position that the state of a classical-mechanical system does. The correct answer to this question was provided by Max Born shortly after Schrödinger discovered the Schrödinger equation. Born postulated that for a one-particle, one-dimensional system, 0 1x, t20 2 dx (1.15) gives the probability at time t of finding the particle in the region of the x axis lying between x and x + dx. In (1.15) the bars denote the absolute value and dx is an infinitesimal length on the x axis. The function 0 1x, t2 0 2 is the probability density for finding the particle at various places on the x axis. (Probability is reviewed in Section 1.6.) For example, suppose that at some particular time t0 the particle is in a state characterized by the wave function ae-bx2 , where a and b are real constants. If we measure the particle’s position at time t0, we might get any value of x, because the probability density a2 e-2bx2 is nonzero everywhere. Values of x in the region around x = 0 are more likely to be found than other values, since 0 0 2 is a maximum at the origin in this case. To relate 0 0 2 to experimental measurements, we would take many identical noninteracting systems, each of which was in the same state . Then the particle’s position in each system is measured. If we had n systems and made n measurements, and if dnx denotes the number of measurements for which we found the particle between x and x + dx, then dnx >n is the probability for finding the particle between x and x + dx. Thus dnx n = 0 0 2 dx and a graph of 11>n2dnx >dx versus x gives the probability density 0 0 2 as a function of x. It might be thought that we could find the probability-density function by taking one system that was in the state and repeatedly measuring the particle’s position. This procedure is wrong because the process of measurement generally changes the state of a system. We saw an example of this in the discussion of the uncertainty principle (Section 1.3). Quantum mechanics is statistical in nature. Knowing the state, we cannot predict the result of a position measurement with certainty; we can only predict the probabilities of various possible results. The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture. Quantum mechanics does not say that an electron is distributed over a large region of space as a wave is distributed. Rather, it is the probability patterns (wave functions) used to describe the electron’s motion that behave like waves and satisfy a wave equation