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VARIATIONAL PRINCIPLES FOR NONSTATIONARY QUANTUM MECHANICAL PROBLEMS AND PERTURBATION THEORY

Yu. N. Demkov

Published Jun 1, 1960• 0 citations
OpenAlex

Abstract

A variational principle is formulated which is invariant with respect to normalization of wave functions and which can be used in the case of nonorthogonal wave functions of the initial and final states. The problera of uniqueness of perturbation theory in this case is considered. The possibility of employing the variational principle for solution of problems concerning decay of the quasi-stationary state and charge exchange is discussed. The main equations for charge exchange at large impact parameters are derived from the variational principle. (auth)

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A variational principle is formulated which is invariant with respect to normalization of wave functions and which can be used in the case of nonorthogonal wave functions of the initial and final states. The problera of uniqueness of perturbation theory in this case is considered. The possibility of employing the variational principle for solution of problems concerning decay of the quasi-stationary state and charge exchange is discussed. The main equations for charge exchange at large impact parameters are derived from the variational principle. (auth)
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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.
Chapter 13 is devoted to two important approximation methods used to solve time-independent problems, namely stationary perturbation theory and the variational method. Perturbation theory is developed for both non-degenerate states and degenerate states, with illustrative examples. The Stark effect is considered separately. The dielectric constant of a transparent medium as a function of frequency is shown to result from the modification of the wave function due to the perturbation of the radiation field. Two complementary topics are presented: the canonical transformation method and the Feynman-Hellman method. Finally, the possibilities of the variational method are illustrated by using it first to determine an upper bound for the energy of the stationary states of a quantum system, and then to study more complicated cases, including the partially shielded Coulomb potential produced by a neutral atom.
The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.
S u m m a r y . A rigorous theory of variational principles in quantum mechanics is formulated. This result is obtained by generalizing the standard theory of the inverse problem of the variational cahmlus, according to new concepts of differentiability on complex Banach spaces. Our theory provides sound mathematical foundations to all known variational principles of quantum mechanics and a powerful tool to find new ones. As a simple application, we derive rigorously two variational principles for the time-dependent SchrSdinger equation: the conventional one, requiring both an initial and a final condition, and a new one, requiring an initial condition only.
The Hamiltonian of a system containing decaying states is subjected to a nonunitary similarity transformation which introduces the wave functions of these decaying states as well as those of the "antidecaying" dual states explicitly into the representation. The dependence of the Green's-function pole on these wave functions is used to formulate a variational principle for their determination, yielding a non-Hermitian, nonlinear wave equation including self-energy contributions. Explicit solutions are given in a simple example.