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.
The Dirac variation-of-constants method has long provided a basis for perturbative solution of the time-dependent Schr\"odinger equation. In spite of its widespread utilization, certain aspects of the method have been discussed only superficially and remain somewhat obscure. The present review attempts to clarify some of these points, particularly those related to secular and normalization terms. Secular terms arise from an over-all time-dependent phase in the wave function, while normalization terms preserve the norm of the wave function. A proper treatment of the secular terms is essential in the presence of a physically significant level shift that can produce secular divergences in the time-dependent perturbation functions. The normalization terms are always important, although the formulation of a simple method for including them is of greatest utility in applications requiring higher-order perturbation theory (e.g., nonlinear optical phenomena), where difficulties have arisen in previous treatments. Although the Dirac perturbation technique includes the correct secular and normalization terms when properly executed, it is convenient to reinterpret the perturbation problem so that the secular and normalization terms can be factored from the wave function to all orders. It is shown that an appropriate over-all multiplicative, time-dependent normalization and phase factor can be obtained, and that it is simply the amplitude for finding the system in the unperturbed eigenstate at any time
t. The regular part of the wave function remaining after this factorization provides a complete description of the physical properties of the system of interest and determines the over-all normalization and phase, as well. Most important, the regular function and its perturbation expansion satisfy equations which are more convenient for computational applications than are the customary Dirac equations, and, in contrast to the latter, they reduce directly to the familiar time-independent perturbation equations in the static limit. To illustrate the general development, the model problem of a linearly perturbed harmonic oscillator and the static, harmonic, and electromagnetic perturbations of arbitrary quantum-mechanical systems are treated explicitly. In the case of an adiabatically applied static perturbation, the familiar adiabatic theorem is recovered with the over-all phase factor giving the perturbed eigenvalue, while in the case of an harmonic perturbation, the overall phase factor obtained includes the system level shift appropriate for a quasiperiodic state. For an electromagnetic perturbation, compact expressions are obtained for various nonlinear optical susceptibilities in forms suitable for computations. Time-dependent Hartree-Fock approximations are treated explicity to demonstrate that difficulties can arise when normalization and secular terms are not extracted prior to application of the perturbation formalism. Connection is also made with other methods which can be employed to eliminate secular and normalization terms from the wave function; these include a projection procedure and multiple-time-scales perturbation theory. The elimination of secular divergences from the perturbation functions is shown to be important for the construction of a valid Fourier transform. Secular and normalization terms also arise in connection with variational principles for the time-dependent Schr\"odinger equation. By employing the Frenkel variational principle and an ansatz for the total wave function that explicitly isolates the secular and normalization terms, a computationally convenient variational functional is obtained. This form of the Frenkel principle provides a bound to the system level shift induced by an oscillatory perturbation and is equivalent to the Ritz variational principle in the static limit. Explicit expressions for the variational functional in the Hartree-Fock approximations are derived in forms suitable for computational applications to the interactions of radiation and matter.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
The reader may gather from Chapters 4 to 6 that the sequence in which we developed the classical formalism of diffusion was to first introduce the equation of motion for dynamical variables. This was achieved through Langevin equations for position and momentum. Diffusive effects were incorporated through heat bath-induced thermal noise and systematic damping parameters. This approach follows the Heisenberg picture of quantum mechanics in which the operators evolve in time whereas the wave function stays constant. Contrast this with the Schrödinger picture in which the wave functions are taken to satisfy the time-dependent Schrödinger equation while the operators remain frozen in time.