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Mathematical Proceedings of the Cambridge Philosophical Society

Time-dependent variational principle and self-consistent field equations

V. U. Nazarov

Published Sep 1, 1985• 5 citations
Publisher

Abstract

In this paper a quantum-mechanical variational principle is proposed, in which the functional varied is the precision, with which the time-dependent Schrödinger equation is satisfied by the wave functions of the given class. Another distinctive feature of our approach is that the independently varied functions are the time derivatives of the one-particle wave functions, of which the wave function of the system is constructed.

DOI
10.1017/s0305004100063544

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In this paper a quantum-mechanical variational principle is proposed, in which the functional varied is the precision, with which the time-dependent Schrödinger equation is satisfied by the wave functions of the given class. Another distinctive feature of our approach is that the independently varied functions are the time derivatives of the one-particle wave functions, of which the wave function of the system is constructed.
The previous time-dependent variation-perturbation theory, which is derived from the conventional time-dependent perturbation equations, contains two inconvenient features. When the perturbation becomes static, the wave function so obtained does not satisfy the time-dependent Schr\"odinger equation, and the perturbation equations cannot be reduced to the ordinary time-independent perturbation equations. To remove these difficulties, we introduce a perturbed time-dependent phase factor in deriving a new set of perturbation equations. We then construct the variation-perturbation functionals as in the previous theory; the perturbed wave functions are solved by minimizing these functionals. The same method is also applied to many-electron systems, using the coupled and uncoupled Hartree-Fock scheme.
The time-independent Schrödinger equation represents a stationary-state equation. The stationary wave functions obtained so far are time independent. Their time-dependence is obtained by means of rather general arguments. Then, stationarystate functions are found. The next step in Newton’s and Euler’s representations of classical mechanics is (to derive) the equation of change of stationary-states. Here, Euler’s principles of stationary-state change are generalized to quantum-mechanical systems. This enables us to derive the quantum-mechanical equation of change of stationary-states. The time-independent Schrödinger equation, i.e, the equation of motion will follow in the next chapter.
Starting with the exact variational formulation of quantum mechanics (or quantum field theory) in the Schroedinger picture, and using Gaussian trail wave functions, an exact functional Schroedinger equation is obtained. The time-dependent Hartree-Fock approximation is obtained and extended to many oscillators and to field theory. The problem of quantum roll, in which an initial Gaussian wave packet starting at an unstable classical equilibrium point evolves in time, is treated using this formalism. Quantum tunnelling is also studied. (LEW)
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.]
Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.
It is shown that the optics-mechanics analogy which originally led Schrödinger to his famous equation stops at the time-independent level. For potentials involving time explicitly, the time-dependent Schrödinger equation cannot be deduced in a similar manner. Instead, a variational principle for the time-dependent Schrödinger equation is established. It minimizes the total quantum fluctuations of a newly defined Hamilton-Jacobi function about its limiting classical value over space-time, thereby demonstrating the existence of a deeper relationship between classical and quantum mechanics beyond the simple optics-mechanics analogy. This principle is similar in spirit to Feynman's space-time approach to quantum mechanics.