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CRC Press eBooks

- Time Dependence

Alastair I. M. Rae, Jim Napolitano

Published Dec 16, 2015• 0 citations
Publisher

Abstract

So far in our development of quantum mechanics, we have given very little consideration to problems involving the time dependence of the wave function. This is rather surprising as in classical mechanics it is time-dependent phenomenai.e., dynamics rather than statics-that command most attention. Moreover, most experimental observations necessarily involve some change in the quantity being observed, so we might expect that the successful prediction of experimental results will require a detailed understanding of the way in which a system changes with time. In fact we have already made some implicit assumptions about time dependence and the principal reason why we have got so far without discussing it in detail is that many observed quantum phenomena are associated with sudden discontinuous changes between otherwise stable states. Thus, most of our information concerning the energy levels of atoms has been obtained from measurements of the frequencies of electromagnetic radiation emitted or absorbed as the atom undergoes a transition from one energy eigenstate to another, assuming the correctness of the formula E = }ω, but not considering the mechanism of the transition in any detail. Assumptions concerning time dependence are also implicit in the quantum theory of measurement which refers to the probability of obtaining a particular result following a measurement performed on a system in a given state; thus the state vector of a spinhalf particle changes from being an eigenvector Sˆ z to being one of Sˆ x following a measurement of the latter property using an appropriately oriented Stern-Gerlach apparatus. However, as was pointed out in Chapter 7, and will be discussed further in Chapters 15 and 16, the collapse of the wave function associated with a measurement is not a direct consequence of the time-dependent Schro¨dinger equation.

DOI
10.1201/b19619-24

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So far in our development of quantum mechanics, we have given very little consideration to problems involving the time dependence of the wave function. This is rather surprising as in classical mechanics it is time-dependent phenomenai.e., dynamics rather than statics-that command most attention. Moreover, most experimental observations necessarily involve some change in the quantity being observed, so we might expect that the successful prediction of experimental results will require a detailed understanding of the way in which a system changes with time. In fact we have already made some implicit assumptions about time dependence and the principal reason why we have got so far without discussing it in detail is that many observed quantum phenomena are associated with sudden discontinuous changes between otherwise stable states. Thus, most of our information concerning the energy levels of atoms has been obtained from measurements of the frequencies of electromagnetic radiation emitted or absorbed as the atom undergoes a transition from one energy eigenstate to another, assuming the correctness of the formula E = }ω, but not considering the mechanism of the transition in any detail. Assumptions concerning time dependence are also implicit in the quantum theory of measurement which refers to the probability of obtaining a particular result following a measurement performed on a system in a given state; thus the state vector of a spinhalf particle changes from being an eigenvector Sˆ z to being one of Sˆ x following a measurement of the latter property using an appropriately oriented Stern-Gerlach apparatus. However, as was pointed out in Chapter 7, and will be discussed further in Chapters 15 and 16, the collapse of the wave function associated with a measurement is not a direct consequence of the time-dependent Schro¨dinger equation.
This chapter aims to introduce some of the applications of the ideas of quantum mechanics that were developed during the last twenty years or so of the twentieth century. The topics chosen reflect our increased understanding of a number of areas and its application to new phenomena that have been predicted and sometimes observed. They all rely on the interplay between the two types of time dependence implicit in the fundamental postulates set out in Chapter 6: Postulate 5 Between measurements, the development of the wave function with time is governed by the time-dependent Schro¨dinger equation.
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
In the previous chapters we have seen how solutions to the time-independent Schro¨dinger equation correspond to the allowed energy levels of a quantum system and how, in the hydrogen atom case in particular, the results of this procedure are in extremely good agreement with the results of experiment. It would be possible to extend the process to predict the energy levels of other atoms. We would find that the corresponding Schro¨dinger equations could no longer be solved exactly, but that approximations could be developed which, when combined with computational techniques, would lead to predicted energy levels that were once again in very good agreement with experiment. However, such a programme, most of which is beyond the scope of this book in any case, would be premature at this stage as we have not yet established a general procedure that will do more than predict the allowed energy levels of a particle moving in a potential and the probability that it is in the vicinity of a particular point in space. We do not yet know, for example, how to predict the momentum of an electron in a hydrogen atom; we do not even know if it has a definite value or if it can only be specified by a probability distribution, as is the case for electron position. We have as yet no way of predicting under what conditions an atom undergoes a transition from one state to another, emitting energy in the form of a quantum of electromagnetic radiation.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
Quantum mechanics has been enormously successful in explaining the behavior of microsystems such as molecules, atoms, and nuclei. We have seen that quantum mechanics describes a quantum system by the wave function ψ, which is the solution of the Schro¨dinger equation. The wave function determines the probability amplitudes (whose modulus squared gives the probability) for all possible outcomes of a measurement corresponding to an observable on the system. In general, the wave function does not uniquely predict the result of a measurement; it only provides a probability distribution for all possible results of the measurement on a quantum system. Only if the system happens to be in an eigenstate corresponding to the observable being measured, is the result of a measurement of this observable completely predictable. On the other hand, the results of measurements corresponding to a conjugate or non-commuting observable, on the same system in the same state, are indeterminate. Quantum mechanical description of physical systems raises profound questions: How are the wave function and the probabilities for different outcomes of a measurement to be interpreted? Does the wave function represent a single system or an ensemble of systems? Does the physical system actually have the attribute corresponding to the measurement in question or does the measurement itself create the attribute? These questions led to different interpretations of quantum mechanics, which were vigorously debated during the years of its discovery as also in the following decades. The new idea of indeterminacy was especially difficult to reconcile with the long-held view that a physical theory should aim at predicting accurately, and without ambiguity, the result of any measurement on any system, big or small. These debates not only gave us a deeper understanding of quantum mechanics but also changed profoundly the way we view the micro-world.