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arXiv:quant-ph/0406062

Stochastic Quantization of the Time-Dependent Harmonic Oscillator

Fernando Haas

Published Jun 9, 2004• 2 citations
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Abstract

We extend the stochastic quantization method recently developed by Haba and Kleinert to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation.
DOI
10.48550/arxiv.quant-ph/0406062

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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.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (x\mathit{x}x , v^\hat{v}v^) , with the commutation relation [x\mathit{x}x , v^\hat{v}v^] = iℏ\mathit{i}\hbariℏ/m\mathit{m}m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K\mathit{K}K(x,v,t\mathit{x, v, t}x,v,t) of the classical system, The comparison with the quantization in the space (x,p\mathit{x, p}x,p) is done with the usual Schrödinger’s equation for the Hamiltonian H\mathit{H}H(x,p,t\mathit{x, p, t}x,p,t), and with the commutation relation [x\mathit{x}x , p^\hat{p}p^​] = iℏ\mathit{i}\hbariℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x\mathit{x}x , v^\hat{v}v^) quantization has fewer oscillations than the (x\mathit{x}x , p^\hat{p}p^​) quantization, the average energy of the system is higher in (x\mathit{x}x , p^\hat{p}p^​) quantization than on the (x\mathit{x}x , v^\hat{v}v^) quantization, and the Boltzmann- Shannon entropy on the (x\mathit{x}x , p^\hat{p}p^​) quantization is higher than on the (x\mathit{x}x , v^\hat{v}v^) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.
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