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CRC Press eBooks

Time-Dependent Perturbation Theory

Gordon Baym

Published Mar 5, 2018• 1 citation
Publisher

Abstract

In the last chapter we studied how to find approximately the energy eigenstates and eigenvalues of a system whose Hamiltonian is of the form H 0 + V, where V is “small” and time independent. Let us now consider a somewhat different situation. Suppose that we begin with a system whose Hamiltonian is H 0 , and then we proceed to act on the system with a time-dependent external force, describable by an interaction term V t added to the Hamiltonian. The question is, what effect does the force have on the system? For example, we might shine light on an atom, and ask what are the chances that the light ionizes the atom. Or we might apply a voltage to a piece of metal, and ask how much current is produced.

DOI
10.1201/9780429499265-12

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In the last chapter we studied how to find approximately the energy eigenstates and eigenvalues of a system whose Hamiltonian is of the form H 0 + V, where V is “small” and time independent. Let us now consider a somewhat different situation. Suppose that we begin with a system whose Hamiltonian is H 0 , and then we proceed to act on the system with a time-dependent external force, describable by an interaction term V t added to the Hamiltonian. The question is, what effect does the force have on the system? For example, we might shine light on an atom, and ask what are the chances that the light ionizes the atom. Or we might apply a voltage to a piece of metal, and ask how much current is produced.
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 many situations in physics, the Hamiltonian of a given system is so complicated that the solution to the stationary Schrodinger equation is practically impossible or very difficult. Therefore, some approximation methods are required. This chapter presents the time-independent perturbation theory, the procedure of finding corrections to non-degenerate eigenvalues and eigenvectors to a part of the Hamiltonian of a given system. The perturbation theory is appropriate when the Hamiltonian can be split into two parts. The perturbation theory can be applied to analyze the quantum properties of particles trapped in two closely coupled potential wells. By closely coupled wells, it is meant that these two wells are separated by a barrier. This is a typical situation in the studies of quantum dynamics of Bose–Einstein condensates. Using the first-order perturbation theory, one can find the eigenvalues and eigenvectors.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
This chapter presents a modernized version of a method originally due to Dirac. This approach depends on a straightforward expansion in powers of the perturbing interaction. The chapter examines a problem where the unperturbed Hamiltonian had a discrete spectrum, and the perturbation was explicitly time dependent. The situation where the spectrum is continuous and the perturbation is independent of time is encountered much more frequently in physics. The chapter explores the derivations of some rather general formulas for collision cross sections. The photo-effect in hydrogen constitutes a very interesting example of a perturbation problem. The dramatic behavior of the matrix element demonstrates that at low energy the Coulomb attraction is very effective in keeping the slow photo-electron near the origin. The chapter examines the result of a collision experiment. It aims to illustrate the power and limitations of the technique, and to establish its relationship to the treatment of elastic scattering.
Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.