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American Journal of Physics

Test of the adiabatic approximation in quantum mechanics: Forced harmonic oscillator

Yukihisa Nogami

Published Jan 1, 1991• 9 citations
Publisher

Abstract

The accuracy of the adiabatic approximation in quantum mechanics is examined by applying it to the forced harmonic oscillator for which the time-dependent Schrödinger equation can be solved exactly. Nonadiabatic transitions caused by the external force that varies in time is focused on. A peculiar situation is illustrated such that, no matter how rapid and large the variation of the external force may be, all nonadiabatic transitions vanish exactly as t→∞.

DOI
10.1119/1.16689

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The accuracy of the adiabatic approximation in quantum mechanics is examined by applying it to the forced harmonic oscillator for which the time-dependent Schrödinger equation can be solved exactly. Nonadiabatic transitions caused by the external force that varies in time is focused on. A peculiar situation is illustrated such that, no matter how rapid and large the variation of the external force may be, all nonadiabatic transitions vanish exactly as t→∞.
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
By invoking the Rabi rotation technique, the Schrödinger wavefunction of a neutral particle with a magnetic moment and a high-spin is exactly solved in a harmonically-changing magnetic field. This result is then applied to illustrate what the accuracy of the quantum adiabatic approximation is exactly and how the non-adiabatic transition vanishes in accompaniment with the appearence of Berry's geometrical phase. The discussion of this paper verifies C. N. Yang's point of view about the generalization of the quantum adiabatic approximation in connection with Berry's phase.