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Acta Physica Sinica (Overseas Edition)

An integral expression of exact solution for harmonic oscillator with time-varying frequency

Li-Xiang Li, Guo Guang-Can

Published Oct 1, 1998• 1 citation
Publisher

Abstract

Using Feynman's method of disentangling operators, an exact expression of evolution operator of harmonic oscillator with time-varying frequency is obtained. The parameters in this expression are related to the time-varying frequency directly. Some implications are further investigated, including general form of linear and quadratic invariants and evolution of Wigner function.

DOI
10.1088/1004-423x/7/10/002

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Using Feynman's method of disentangling operators, an exact expression of evolution operator of harmonic oscillator with time-varying frequency is obtained. The parameters in this expression are related to the time-varying frequency directly. Some implications are further investigated, including general form of linear and quadratic invariants and evolution of Wigner function.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For simplicity, we consider a free particle with a variable mass M(t), a particle with a variable mass M(t) in an alternating homogeneous field, and a harmonic oscillator with a variable mass M(t) and frequency ω(t) subject to a variable force F(t). To construct the evolution operators for these systems in an explicit disentangled form, we use a simple technique to find the general solution of a certain class of differential and finite-difference nonstationary Schrödinger-type equations of motion and also the operator identities of the Baker–Campbell–Hausdorff type. With known evolution operators, we can easily find the most general form of the propagators, invariants of any order, and wave functions and establish a unitary relation between systems. Results known in the literature follow from the obtained general results as particular cases.
For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique of integration within an ordered product of operators. The normally ordered time evolution operator is thus obtained. We then derive the Wigner function of u(t)|n⟩, where |n⟩ is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.