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Journal of Mathematical Physics

Exponential time-evolution operator for the time-dependent harmonic oscillator

Francisco Marcelo Fernández

Published Dec 1, 1987• 16 citations
Publisher

Abstract

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

DOI
10.1063/1.527819

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The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.
For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.