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American Journal of Physics

Time-dependent perturbation and exact results for the periodically driven quantum harmonic oscillator

Russell Akridge

Published Feb 1, 1995• 5 citations
Publisher

Abstract

Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.

DOI
10.1119/1.17971

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Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.
This paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity. We describe the effective evolution equation of the state vector, which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.
As an application of the recently proposed Langevin formulation of quantum dynamics, we quantize the time-dependent harmonic oscillator. In this way, we are able to explicitly compute the quantum-mechanical propagator. At variance with previous results, the expression found in this letter contains no auxiliary function to be determined by solving (as yet unsolved) non-linear differential equations, but it is written in terms of two series which arise quite naturally within the present formalism. Our strategy is compared with ordinary time-dependent perturbation theory.
We use the Lewis and Riesenfeld invariant method [J. Math. Phys. 10, 1458 (1969)]10.1063/1.1664991 and a unitary transformation to obtain the exact Schrödinger wave functions for time-dependent harmonic oscillators exhibiting log-periodic-type behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with m = m0t/t0 and ω = ω0t0/t, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with m and ω constant.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.