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CRC Press eBooks

Time-Dependent Systems

Walter Wilcox

Published Aug 23, 2019• 0 citations
Publisher

Abstract

Synopsis: The time development of quantum systems with time-dependent Hamiltonians or potentials will be discussed in this chapter. First, we set up the formalism of time development in a very general way and discover that there are two basic Hilbert space descriptions of these processes. As discussed earlier, exact solutions for physical systems are rare, and we therefore find approximate ways of describing fast and slow time developments. Concentrating on two-state systems, we investigate the important role complex phases play in the slow evolution of quantum systems. In this context, we learn how to apply perturbation theory to our formalism, resulting in wonderfully practical formulas that can readily be applied in the lab, including elastic scattering and spontaneous emission rates. Finally, the important topic of time decay of systems and associated uncertainties in the energies of such states will be discussed.

DOI
10.1201/b22285-10

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Synopsis: The time development of quantum systems with time-dependent Hamiltonians or potentials will be discussed in this chapter. First, we set up the formalism of time development in a very general way and discover that there are two basic Hilbert space descriptions of these processes. As discussed earlier, exact solutions for physical systems are rare, and we therefore find approximate ways of describing fast and slow time developments. Concentrating on two-state systems, we investigate the important role complex phases play in the slow evolution of quantum systems. In this context, we learn how to apply perturbation theory to our formalism, resulting in wonderfully practical formulas that can readily be applied in the lab, including elastic scattering and spontaneous emission rates. Finally, the important topic of time decay of systems and associated uncertainties in the energies of such states will be discussed.
Abstract Quantum perturbation is an important tool for developing approximate solutions to quantum problems that cannot be solved exactly. This chapter provides a brief, systematic development of both time-independent and time-dependent quantum perturbation theory and the physical interpretation of its results, including both first- and second-order corrections. A quartic perturbation of the simple harmonic oscillator is used to illustrate time-independent perturbation theory. A spin-1/2 system subject to an oscillating field is used in time-dependent perturbation theory to describe the absorption and emission of energy from the quantum system.
How to deal with time-dependent potentials, what kinds of physical problems having time-dependent potentials can be solved exactly and which are to be solved with the use of perturbation theory, what are different possible forms of time-dependent perturbating fields, and semi-classical treatment of the interaction of fields with matter are the subject matters of this chapter. First order perturbation theory, computation of transition probability, transition rates and Fermi's Golden Rule for harmonic and constant perturbations, absorption, stimulated emission and dipole approximation when electromagnetic fields interact with matter, spontaneous emission and Einstein coefficients, and dipole selection rules are other topics which are dealt with in detail in the chapter. Discussions with properly justified mathematical treatment on time-dependent potentials and on the exactly solvable time-dependent two-state systems are the special foci of this chapter.
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.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.
Abstract In this chapter we develop methods, both exact and approximate, for timedependent problems (Adler, 1988), which complement the methods for stationary state problems given in Chapter 5 and their application to scattering theory in Chapter 6. We begin by giving the general equations for time-dependent perturbation theory in quaternionic quantum mechanics. By making an appropriate choice of initial conditions, these equations are applied successively to the problems of scattering theory and of decaying state theory. We next discuss the use of the interaction and Heisenberg pictures, particularly in the context of the quaternionic forced harmonic oscillator. We conclude by briefly discussing the use of quaternionic quantum mechanics as a model for time reversal violation in elementary particle physics (Adler, l 986a,b).
Chapter 17 begins by approaching the general problem of a quantum system subject to a time-dependent perturbation. We then apply perturbation theory to the interaction of atomic matter with the radiation field and derive the most important radiative corrections. Atomic transitions – both induced and spontaneous – are discussed in detail, with some important applications. The formal introduction of field quantization allows us to make contact with quantum electrodynamics and quantum optics and discuss the Jaynes–Cummings model. Stochastic electrodynamics, in which the matter–field interaction is not a perturbation but is present ab initio, is used to explain atomic stability and its breakdown leading to radiative transitions. In the same spirit, the quantization of the field is physically justified by deriving the conventionally postulated field operators and their basic commutators. The chapter ends with a brief introduction to second quantization, an elegant and powerful formalism for describing and analyzing quantum many-body systems.