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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Energy distribution of the quantum harmonic oscillator under random time-dependent perturbations

Josselin C. Garnier

Published Oct 1, 1999• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.60.3676

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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.
This paper investigates the evolution of a quantum particle in a harmonic oscillator whose spring coefficient randomly fluctuates around its mean value. 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 which tends to zero and a perturbation duration which 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.
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.
The behavior of a three-dimensional, nonrelativistic, quantum mechanical harmonic oscillator is investigated under the influence of three distinct types of randomly fluctuating potential fields. Specifically, kinetic (or transport) equations are derived for the corresponding stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution density function) and the stochastic Liouville equation (correspondence limit approximation) using two closely related statistical techniques, the first-order smoothing and the long-time Markovian approximations. Several physically important averaged observables are calculated in special cases. In the absence of a deterministic inhomogeneous potential field (randomly perturbed, freely propagating particle), the results reduce to those reported previously by Besieris and Tappert.
The Hamiltonian quantum theory of the damped harmonic oscillator is investigated by using an explicitly time-dependent Hamiltonian including a commutative stochastic force. The propagator is computed and shown to describe an inhomogeneous quantum dynamical semigroup evolution on the Banach cone of density matrices. The moments tend to the thermal equilibrium values independently of the initial conditions, when time tends to infinity and some rescaling procedure is performed. With respect to the rescaling neither the Heisenberg nor the Schrodinger picture exists. Furthermore the energy dissipates below the zero-point energy in violation of the Heisenberg uncertainty relation.
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.
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.