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Chinese Journal of Physics

Time-Dependent Variational Principle

Der-Ruenn Su

Published Oct 1, 1967• 1 citation
OpenAlex

Abstract

Variational principles for quantum theory of scattering are given. Four variational principles give the matrix elements of transition operator and two variational principles give the matrix elements of reaction operator. All the forms of the variational principles are similiar to the Schwinger variational principle for the scattering amplitude in the stationary collision theory.

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Variational principles for quantum theory of scattering are given. Four variational principles give the matrix elements of transition operator and two variational principles give the matrix elements of reaction operator. All the forms of the variational principles are similiar to the Schwinger variational principle for the scattering amplitude in the stationary collision theory.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
Variational principles are designed for the solution of the Schr\"odinger equation when a point source is placed in the presence of an inhomogeneous, absorbing medium represented by an arbitrary complex potential function. When the point source is allowed to recede to infinity, these stationary structures reduce to variational principles for the wave function in the standard scattering problem, namely the outgoing solution to the Schr\"odinger equation for an incident plane wave. Finally in the asymptotic region, the well-known bifunctional variational principles for the transition amplitudes arise automatically from the stationary forms for the wave function describing the standard scattering problem. A few examples leading to variationally improved wave functions are discussed.
The time-dependent variational principle of the Schr\"odinger equation is applied to a formally exact solution of the Schr\"odinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schr\"odinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]