Modern. Quantum Mechanics. Revised Edition. J. J. Sakurai. Late, University of California, Los Angeles. San Fu Tuan, Editor. University of Hawaii, Manoa. A. S 2S 2 o f (A~^b MODERN QUANTUM MECHANICS Second Edition s Addison- Wesley Boston Columbus Indianapolis New York San Francisco UpperSaddle. Book: Modern Quantum Mechanics Introduction to Quantum Mechanics by David J Griffiths. maroc-evasion.info
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The formal solution to 2. The Tf s at different times do not commute. Continuing with the ex- ample involving spin-magnetic moment, we suppose, this time, that the magnetic held direction also changes with time: The formal solution in such a situation is given by I dt' Hit' Jtn 2.
Dyson, who developed a perturbation expansion of this form in quantum held theory. We do not prove 2. In elementary applications, only case 1 is of practical interest. In the remaining part of this chapter we assume that the H operator is time-independent.
We will encounter time-dependent Hamiltonians in Chapter 5. Energy Eigenkets To be able to evaluate the effect of the time-evolution operator 2. This is particularly straightforward if the base kets used are eigenkets of A such that [A,H] — 0; 2. Notice that the relative phases among various com- ponents do vary with time because the oscillation frequencies are different. It is in this sense that an observable compatible with H [see 2. We will encounter this connection once again in a different form when we discuss the Heisenberg equation of motion.
In the foregoing discussion the basic task in quantum dynamics is reduced to finding an observable that commutes with H and evaluating its eigenvalues. Once that is done, we expand the initial ket in terms of the eigenkets of that observ- able and just apply the time-evolution operator.
This last step amounts merely to changingthe phase of each expansion coefficient, as indicatedby 2. Even though we worked out the case where there is just one observable A that commutes with H, our considerations can easily be generalized when there are several mutually compatible observables all also commuting with H: It is therefore of fundamental importance to find a complete set of mutually compatible observ- ables that also commute with H. Once such a set is found, we express the initial ket as a superposition of the simultaneous eigenkets of A,B, C, The final step is just to apply the time-evolution operator, written as 2.
In this manner we can solve the most general initial-value problem with a time-independent H. Time Dependence of Expectation Values It is instructive to study how the expectation value of an observable changes as a function of time. Suppose that at t — 0 the initial State is one of the eigenstates of an observable A that commutes with H, as in 2. We now look at the expectation value of some other observable B, which need not commute with A or with H.
So the expectation value of an observable taken with respect to an energy eigenstate does not change with time. For this reason an energy eigenstate is often ref erred to as a stationary State.
The situation is more interesting when the expectation value is taken with re- spect to a superposition of energy eigenstates, or a nonstationary State. W e consider an extremely simple system that, however, illustrates the basic formalism we have developed. Furthermore, we take B to be a static, uniform magnetic held in the z-direction. The sum of the two probabilities is seen to be unity at all times, in agreement with the unitarity property of the time-evolution operator.
We will comment further on spin precession when we discuss rotation operators in Chapter 3. Experimentally, spin precession is well established. In faet, it is used as a tool for other investigations of fundamental quantum-mechanical phenomena. For ex- ample, the form of the Hamiltonian 2. However, higher-order corrections from quantum held theory predict a small but precisely calculable deviation from this, and it is a high priority to produce competitively precise measurements of g Such an experiment has been recently completed.
See G. Bennett et al. D 73 Consequently, observation of their precession measures g — 2 direetly, facilitating a very precise result.
Figure 2. They determine t: The size of the signal decreases with time because the muons decay. Neutrino Oscillations A lovely example of quantum-mechanical dynamics leading to interference in a two-state system, based on current physics research, is provided by the phe- nomenon known as neutrino oscillations.
Neutrinos are elementary particles with no charge and very small mass, much smaller than that of an electron. These are in faet eigenstates of a Hamiltonian that Controls those interactions. On the otherhand, it is possible and, in faet, is now known to be true that neu- trinos have some other interactions, in which case their energy eigenvalues cor- respond to States that have a well-defined mass. Indeed, there is no strong theoretical bias for any particular value of 9, and it is a free parameter that, today, can be determined only through experiment.
Neutrino oscillation is the phenomenon by which we can measure the mixing angle. Then according to 2. If the difference in the masses is small enough, then this phase difference can build up over a macroscopic distance. In faet, by measuring the interference as a funetion of difference, one can observe oscillations with a period that depends on the difference of masses, and an amplitude that depends on the mixing angle.
It is straigh tforward see Problem 2. In this case, the Hamiltonian is just that for a free particle, but we need to take some care. Neutrinos are very low mass, so they are highly relativistic for any practical experimental conditions. Therefore, for a fixed momentum p, the energy eigen value for a neutrino of mass m is given to an extremely good approximation as 78 Chapter 2 Quantum Dynamics FIGURE 2.
Abe et al. The oscillations predicted by 2. See Figure 2. The curve is not a perf eet sine wave because the reactors are not all at the same distance from the detector.
Correlation Amplitude and the Energy-Time Uncertainty Relation We conclude this section by asking how State kets at different times are correlated with each other. To estimate 2. Expression 2. Writing 2. In Chapter 5 we will come back to 2.
There is another formulation of quantum dynamics where observables, rather than State kets, vary with time; this second approach is known as the Heisenberg picture.
Before discussing the dif- ferences between the two approaches in detail, we digress to make some general comments on unitary operators. Unitary operators are used for many different purposes in quantum mechan- ics. In this book we introduced Section 1. In that section we were concerned with the question of how the base kets in one representation are related to those in some other representations.
The State kets themselves are assumed not to change as we switch to a different set of base kets, even though the numerical values of the expansion coefficients for la are, of course, different in different representations. Subsequently we introduced two unitary operators that actually change the State kets, the translation operator of Section 1. This mathematical identity suggests two approaches to unitary transformations: Approach 1: This is possible because these operations actually change quantities such as x andL, which are observables of classical mechanics.
We therefore conjecture that a doser connection with classical physics may be established if we follow approach 2. A simple example may be helpful here. We go back to the infinitesimal transla- tion operator T dx '. The formalism presented in Section 1. In the previous section we examined how State kets evolve with time.
This means that we were following approach 1 , which is known as the Schrodinger picture when it is applied to time evolution. Alternatively, we may follow approach 2, known as the Heisenberg picture when applied to time evolution. In the Schrodinger picture the operators corresponding to observables such as x, p y , and S z are fixed in time, while State kets vary with time, as indicated in the previous section.
In contrast, in the Heisenberg picture the operators correspond- ing to observables vary with time; the State kets are fixed — frozen, so to speak — at what they were at to. It is convenient to set to in V. Assuming that does not depend explicitly on time, which is the case in most 2.
Indeed, historically 2. Dirac, who — with his characteristic modesty — called it the Heisenberg equation of motion. It is worth noting, however, that 2. For example, the spin operator in the Heisenberg picture satisfies dt 2.
With this assumption we can reproduce the correct classical equations in the classical limit. When the physical system in question has no classical analogues, we can only guess the structure of the Hamiltonian operator. We try various forms un til we get the Hamiltonian that leads to results agreeing with empirical observation. In practical applications it is often necessary to evaluate the commutator of Xi or pt with functions of xj and pj.
To this end thefollowing formulas are useful: We can easily proveboth formulas by repeatedly applying 1. We are now in a position to apply the Heisenberg equation of motion to a free particle of mass m. The Hamiltonian is taken to be of the same form as in classical mechanics: Quite generally, it is evident from the Heisenberg equation of motion 2. We now add a potential V x to our earlier free-particle Hamiltonian: Using 2.
Ehrenfest, who derived it in 1 using the formalism of wave mechanics. When the theorem is written in this expectation form, its validity is independent of whether we are using the Heisenberg or the Schrodinger picture; after all, the expectation values are the same in the two pictures. In contrast, the operator form 2. We note that in 2. It is therefore not surprising that the center of a wave packet moves like a classical particle subjected to V x. Base Kets and Transition Amplitudes So far we have avoided asking how the base kets evolve with time.
A common misconception is that as time goes on, all kets move in the Schrodinger picture and are stationary in the Heisenberg picture. This is not the case, as we will make clear shortly. The important point is to distinguish the behavior of State kets from that of base kets.
In the Schrodinger picture, A does not change, so the base kets, ob- tained as the solutions to this eigenvalue equation at t — 0, for instance, must re- main unchanged. Unlike State kets, the base kets do not change in the Schrodinger picture. As time goes on, the Heisenberg-picture base kets, denoted by a',t n, move as follows: This is consistent with the theorem on unitary equivalent ob- servables discussed in Section 1.
We see that the expansion coefficients ofa State ket in terms of base kets are the same in both pictures: These considerations apply equally well to base kets that exhibit a continuous Spectrum; in particular, the wave function x 7 la can be regarded either as 1 the inner product of the stationary position eigenbra with the moving State ket the Schrodinger picture or as 2 the inner product of the moving position eigenbra with the stationary State ket the Heisenberg picture.
We will discuss the time dependence of the wave function in Section 2. To illustrate further the equivalence between the two pictures, we study transi- tion amplitudes, which will play a fundamental role in Section 2. At some la ter time t we may ask, What is the probability amplitude, known as the transition amplitude, for the system to be found in an eigenstate of observable B with eigenvalue b'l Here A and B can be the same or different. In the Schrodinger picture the State ket at t is given by V.
In contrast, in the Heisenberg picture the State ket is stationary — that is, it remains as a' at all times — but the base kets evolve oppositely. To conclude this section, let us summarize the differences between the Schrodinger picture and the Heisenberg picture.
Table 2. It not only illustrates many of the basic concepts and methods of quan- tum mechanics but also has much practical value.
Essentially any potential well can be approximated by a simple harmonic oscillator, so it describes phenomena from molecular vibrations to nuclear structure. Moreover, because the Hamilto- nian is basically the sum of squares of two canonically conj ugate variables, it is also an important starting point for much of quantum held theory. Born and N. Wiener, to obtain the energy eigenkets and energy eigenvalues of the simple harmonic oscillator.
The basic Hamiltonian is 9 2 mco L x 2. The operators x and p are, of course, Hermitian. Because of 2. Because the increase decrease of n by one amounts to the creation annihilation of one quantum unit of energy hco, the term creation operator annihilation operator for a ' a is deemed appropriate.
Equation 2. One may argue that if we start with a noninteger n, the sequence will not terminate, leading to eigenkets with a negative value of n. This is not surprising because x and p, like a and a ] , do not commute with N. The operator method can also be used to obtain the energy eigenfunctions in position space. We see that the to 2. From 2. In contrast, the uncertainty products for the excited States are larger: Time Development of the Oscillator So far we have not discussed the time evolution of oscillator State kets or of ob- servables such as x and p.
The Heisenberg equations of motion forp and x are, from 2. In terms of x and p, we can rewrite 2. For pedagogical reasons, we now present an alternative derivation of 2. We leave the proof of this formula, which is known as the Baker-Hausdorff lemma, as an exercise. Applying this formula to 2. However, this inference is not correct. This point is also obvious from our earlier conclusion see Section 2.
We have seen that an energy eigenstate does not behave like the classical oscillator — in the sense of oscillating expectation values for x and p — no matter how large n may be. We may logically ask, How can we construct a superposition of energy eigenstates that most closely imitates the classical oscillator? In wave- function language, we want a wave packet that bounces back and forth without spreading in shape.
The coherent State has many other remarkable properties: It can be obtained by translating the oscillator ground State by some finite distance. It satisfies the minimum uncertainty product relation at all times. A systematic study of coherent States, pioneered by R. Glauber, is very rewarding; the reader is urged to work out Problem 2. The Hamiltonian operator is taken to be H — E V x.
Later in this book we will consider more- complicated Hamiltonians — a time-dependent potential V x,t ; a nonlocal but separable potential where the right-hand side of 2. We first write the Schrodinger equation for a State ket 2. See also the discussion on squeezed light at the end of Section 7. This equation is, in faet, the starting point of many textbooks on quantum mechanics. In our formalism, however, this is just the Schrodinger equation for a State ket written explicitly in the x-basis when the Hamiltonian operator is taken to be 2.
The Time-lndependent Wave Equation We now derive the partial differential equation satisfied by energy eigenfunetions. We showed in Section 2. Let us now substitute 2. Actually, in wave mechanics where the Hamiltonian operator is given as a funetion of x and p, as in 2. We may therefore omit reference to a! Schrodinger — announced in the first of four monumental papers, all written in the first half of — that laid the foundations of wave mechanics.
In the same paper he immediately applied 2. To solve 2. Suppose we seek a solution to 2. We know from the theory of partial differential equations that 2. It is in this sense that the time-independent Schrodinger equation 2. In both cases we solve boundary- value problems in mathematical physics.
A short digression on the history of quantum mechanics is in order here. The faet that exaetly soluble eigenvalue problems in the theory of partial differential equations can also be treated using matrix methods was already known to math- ematicians in the first quarter of the twentieth century.
Furthermore, theoretical physicists like M. Born frequently consulted great mathematicians of the day — D. Hilbert and H. Weyl, in particular. Yet when matrix mechanics was born in the summer of , it did not immediately occur to the theoretical physicists or the mathematicians to reformulate it using the language of partial differential equations.
However, a close inspection of his papers shows that he was not at all influenced by the earlier works of Heisenberg, Born, and Jordan. Instead, the train of reasoning that led Schrodinger to formulate wave mechanics has its roots in W. Once wave mechanics was formulated, many people, including Schrodinger himself, showed the equivalence between wave mechanics and matrix mechanics.
It is assumed that the reader of this book has some experience in solving the time-dependent and time-independent wave equations. In this book, we do not thoroughly cover these more elementary topics and so- lutions.
Some of these for example, the harmonic oscillator and hydrogen atom are pursued, but at a mathematical level somewhat higher than what is usually seen in undergraduate courses. Interpretations of the Wave Function We now turn to discussions of the physical interpretations of the wave function.
Specifically, when we use a detector that ascertains the presence of the particle within a small volume element d 3 x' around x', the probability of recording a positive result at time t is given by p x',t d 3 x'. In the remainder of this section we use x for x' because the position operator will not appear.
Conversely, a complex potential can phenomenologically account for the disappearance of a particle; such a potential is often used for nuclear reactions where incident particles get absorbed by nuclei. This is indeed the case for j int egrat ed over all space.
If we adopt such a view, we are led to face some bizarre consequences. A typical argument for a position measurement might go as follows. An atomic electron is to be regarded as a continuous distribution of matter Alling up a finite region of space around the nucleus; yet, when a measurement is made to make sure that the electron is at some particular point, this continuous distribution of matter suddenly shrinks to a point-like particle with no spatial extension.
The meaning of p has already been given. The direction of j at some point x is seen to be normal to the surface of a constant phase that goes through that point. However, we would like to caution the reader against too literal an interpretation of j as p times the velocity defined at every point in space, because a simultaneous precision measurement of position and velocity would necessarily violate the uncertainty principle.
The Classical Limit We now discuss the classical limit of wave mechanics. Straigh tforward differentiations lead to!
Let us supposenow that h can, in some sense, be regarded as a small quantity. We can then collect terms in 2. We have a semiclassical interpretation of the phase of the wave function: In classical mechanics the velocity vector is tangential to the particle trajectory, and as a result we can trace the trajectory by following continuously the direction of the velocity vector.
The particle trajectory is like a ray in geometric optics because the V S that traces the trajectory is normal to the wave front defined by a constant S. In this sense, geometrical optics is to wave optics what classical mechanics is to wave mechanics. One might wonder, in hindsight, why this optical-mechanical analogy was not fully exploited in the nineteenth century.
Besides, the basic unit of action h, which must enter into 2. The solution in spherical coordinates will be left until our treatment of angular momentum is presented in the next chapter. Differential equation 2. We impose periodic boundary conditions on the box and thereby obtain a finite normalization constant C. That is, 2n 2 tt 2tz 2. In faet, in the realistic limit where L is very large, there can be a large 2.
See, for example, the discussion of the photoelectric effect in Section 5. Rewriting 2. First, transform 2. Typically, one would now look for a se- ries solution for h y and discover that a normalizable solution is possible only if the series terminates.
In faet, we use this approach for the three-dimensional isotropic harmonic oscillator in this book. See Section 3. Let us take a different approach. We can take derivatives of g x,t to build the Hermite polynomials using re- cursion relations between them and their derivatives. The trick is that we can differentiate the analytic form of the generating function 2. For example, if we take the derivative using 2.
This is enough information for us build the Hermite polynomials: To see why it is relevant to the simple harmonic oscillator, consider the derivative of the generating function with respect to t.
If we start with 2. This, however, is the same as the Schrodinger equation written as 2. See Problem 2. Generating functions have a usefulness that far outreaches our limited appli- cation here.
Among other things, many of the orthogonal polynomials that arise from solving the Schrodinger equation for different potentials can be derived from generating functions. See, for example, Problem 3. The interested reader is encouraged to pursue this further from any one of the many excellent texts on mathematical physics.
This point will be important for understanding the quantum behavior of a particle of mass m bound by this potential. Once again, we write the differential equation in terms of dimensionless vari- ables, based on appropriate scales for length and energy. The Airy function has a peculiar behavior, oscillatory for negative values of the argument and decreasing rapidly toward zero for positive values. Of course, this is exactly the behavior we expect for the wave function, since z — 0 is the classical tuming point.
In other words, the zeros of the Airy function or its derivative determine the quantized energies. The quantum-theoretical treatment of the linear potential may appear to have little to do with the real world. It turns out, however, that a potential of type 2. In this case, the x in 2. The bouncing hall happens to be one of those rare cases where quantum- mechanical effects can be observed macroscopically.
Plotted is the detected neutron rate as a function of the height of a slit that allows neutrons to pass only if they exceed this height. The WKB Semiclassical Approximation Having solved the problem of a linear potential, it is worthwhile to introduce an important approximation technique known as the WKB solution, after G.
Wentzel, A. Kramers, and L.
Nesvizhevsky et al. D 67 , and V. C 40 D 67 The solid curve is a fit to the data based on classical physics. Note that the vertical scale is loga- rithmic. Such is never the case near classical tuming points, but this is where the linear potential solution can be used to join the solutions on either side of them. In this case, 2. However, we consider a solution to this equation under the condition that ti d 2 W dx 2 2. Forging ahead for now, we use the condition 2.
Note the similarity to Figure 2. Joining these two solutions across the classical turning point is the next task. We do not discuss this joining procedure in detail, because it is discussed in many places Schiff , pp.
Instead, we content ourselves with presenting the results of such an analysis for a potential well, schematically shown in Figure 2. The wave function must behave like 2. The solutions in the neighbor- hood of the turning points, shown as a dashed line in Figure 2. It is relevant to this discussion, but we are glossing over the details. This implies that the arguments of the cosine in 2.
In this way we obtain a very interesting consistency condition: Sommerfeld and W. One might be tempted to use 2. We note, however, that 2. This integral is elementary, and we obtain 2.
Before concluding, let us return to the interpretation of the condition 2. In faet, when we use 2. Indeed, condition 2.
We also note thatcondition 2. Roughly speaking, the potential must be essentially constant over many wavelengths. Thus we see that the semiclassical picture is reliable in the short-wavelength limit. Let us translate this statement into the language of wave mechanics. Note also that 2. Now 2. It can be constructed once the energy eigen- functions and their eigenvalues are given. The only peculiar feature, if any, is that when a measurement intervenes, the wave function changes abruptly, in an uncontrollable way, into one of the eigenfunctions of the observ- able being measured.
There are two properties of the propagator worth recording here. This is evident from 2. Because of these two properties, the propagator 2. Indeed, this interpretation follows, perhaps more elegantly, from noting that 2. In this manner we can add the various contributions from different positions x'.
This situation is analogous to one in electrostatics; if we wish to find the electrostatic potential due to a general charge distribution p x' , we first solve the point-charge problem, multiply the point- charge solution by the charge distribution, and integrate: The delta function 8 t — to is needed on the right-hand side of 2. The particular form of the propagator is, of course, dependent on the particular potential to which the particle is subjected.
Consider, as an example, a free particle in one dimension. Here we simply record the result: It can also be obtained using the a, a" operator method Saxon , pp. Notice that 2. This means, among other things, that a particle initially localized precisely at x! Now we see that 2. Next, let us consider the Laplace-Fourier transform of G t: But we can make the integral meaning- ful by letting E acquire a small positive imaginary part: Propagator as a Transition Amplitude To gain further insight into the physical meaning of the propagator, we wish to relate it to the concept of transition amplitudes introduced in Section 2.
Likewise, the 2. In Section 2. Because there is nothing special about the choice of to — only the time difference t — to is relevant — we can identify x", t x', to as the probability amplitude for the particle prepared at to with position eigenvalue x' to be found at a later time t at x".
Roughly speaking, x",t x',to is the amplitude for the particle to go from a space-time point x',to to another space-time point x",t , so the term transition amplitude forthis expression is quite appropriate. This interpretation is, of course, in complete accord with the interpretation we gave earlierfor K x",p,x',to. Yet another way to interpret x",t x',to is as follows. Because at any given time the Heisenberg-picture eigenkets of an observable can be chosen as base kets, we can regard x",t x',to as the transfor- mation function that connects the two sets of base kets at dijferent times.
This is reminiscent of classical physics, in which the time development of a classical dynamic variable such as x t is viewed as a canonical or contact transformation generated by the classical Hamiltonian Goldstein , pp.
It turns out to be convenient to use a notation that treats the space and time coordinates more symmetrically. This kind of reasoning leads to an independent formulation of quantum mechanics that R. R Feynman published in , to which we now turn our attention. Path Integrals as the Sum Over Paths Without loss of generality we restrict ourselves to one-dimensional problems.
Before proceeding further, it is profitable to review here how paths appear in classical mechanics. On the contrary, there exists a unique path that corresponds to the actual motion of the classical particle. Feynman's Formulation The basic difference between classical mechanics and quantum mechanics should now be apparent. How are we to accomplish this? As a young graduate student at Princeton University, R.
Feynman tried to attack this problem. Feynman attempted to make sense out of this remark. In so doing he was led to formulate a space-time approach to quantum mechanics based on path integrals. For compactness, we introduce a new notation: So even though the path dependence is not explicit in this notation, it is understood that we are considering a particular path in evaluating the integral.
Imagine now that we are following some prescribed path. We must still integrate overx 2 ,X 3 ,. Before presenting a more precise formulation, let us see whether considera- tions along this line make sense in the classical limit. So most paths do not contribute when ft is regarded as a small quantity. However, there is an important exception. We denote the S that satisfies 2.
We now attempt to deform the path a little bit from the classical path. The resulting S is still equal to S m in to first order in deformation. As a result, as long as we stay near the classical path, constructive interferencebetween neighboring paths is possible. The reader may work out a similar com- parison for the simple harmonic oscillator.
Ju tN , T c Iassical x,x dt 2. Its meaning as the sum over all possible paths should be apparent from 2. Our steps leading to 2. As a Chegg Study subscriber, you can view available interactive solutions manuals for each of your classes for one low monthly price.
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