Eigenvalue Problem for Schrödinger Operators and Time-Dependent Harmonic Oscillator

It is shown that the eigenvalue problem for the Hamiltonians of the standard form, H = p/(2m) + V (x), is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the solution of the dynamical equations of a class of time-dependent harmonic oscillators is also pointed out. E-mail: [email protected] Address after July 1, 1997 1 Recently there has been a growing interest in the study of the quantum dynamics, i.e., solution of the Schrödinger equation Hψ = ih̄ dψ dt , (1) for time-dependent harmonic oscillator [1], H = p 2M(t) + 1 2 M(t)ω(t)x , (2) and its generalizations [2]. The basic idea used in these studies is the invariant method of Lewis and Riesenfeld [3]. Although the results of Refs. [1, 2] have direct relevance for the construction of the squeezed states which have potential physical applications, contrary to the claims made by the authors, they do not yield exact solution of the Schrödinger equation (1). In reality, what is being done [1] is to show that the general solution of the Schrödinger equation can be expressed in terms of the solutions of the classical equation of motion. This is a second order differential equation with variable coefficients whose exact solution is not known. The purpose of this article is to show that an exact solution of the dynamical equation for time-dependent harmonic oscillator includes as a special case the solution for the eigenvalue problem for arbitrary time-independent Hamiltonians of the standard form H = p/(2m) + V (x), i.e.,