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Journal of Mathematical Physics

An alternative derivation of transition amplitudes for time-dependent harmonic oscillators

H. E. De Meyer, Guido Vanden Berghe

Published Jul 1, 1981• 3 citations
Publisher

Abstract

It is shown that the relations which link the x and p operators of a time-dependent harmonic oscillator to the x and p operators of the corresponding time-independent oscillator can be cast into a generalized Bogoliubov transformation on the standard harmonic oscillator boson creation and annihilation operators. Once this fact has been recognized, various techniques can be invoked to derive in a short way transition amplitudes.

DOI
10.1063/1.525064

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It is shown that the relations which link the x and p operators of a time-dependent harmonic oscillator to the x and p operators of the corresponding time-independent oscillator can be cast into a generalized Bogoliubov transformation on the standard harmonic oscillator boson creation and annihilation operators. Once this fact has been recognized, various techniques can be invoked to derive in a short way transition amplitudes.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.
The unitary operator V ( t ) that diagonalizes the time-dependent quadratic Hamiltonian (TDQH) into a time-dependent harmonic oscillator (TDHO) is obtained using a Lie algebra. The method involves a factorization of the TDQH into a TDHO through a unitary Bogoliubov transformation in terms of creation and annihilation operators with time-dependent coefficients. It is shown that this operator can be easily achieved by means of the factorization, together with the commonly known Wei–Norman theorem. We discuss the conditions under which this unitary operator converges to the evolution operator U ( t ) of the Schrödinger equation for the TDQH, giving then a straightforward calculation of the evolution operator with respect to the procedures published in the literature.