Phase space path integral approach to harmonic oscillator with a time-dependent force constant

Janakiraman, Deepika and Sebastian, KL (2015) Phase space path integral approach to harmonic oscillator with a time-dependent force constant. In: PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS, 433 . pp. 198-203.

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Abstract

The quantum statistical mechanical propagator for a harmonic oscillator with a time-dependent force constant, m omega(2)(t), has been investigated in the past and was found to have only a formal solution in terms of the solutions of certain ordinary differential equations. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical expressions for such path integrals is of great interest. In a previous work, we had obtained the exact propagator for motion in an arbitrary time-dependent harmonic potential in the overdamped limit of friction using phase space path integrals in the context of Levy flights - a result that can be easily extended to Brownian motion. In this paper, we make a connection between the overdamped Brownian motion and the imaginary time propagator of quantum mechanics and thereby get yet another way to evaluate the latter exactly. We find that explicit analytic solution for the quantum statistical mechanical propagator can be written when the time-dependent force constant has the form omega(2)(t) = lambda(2)(t) - d lambda(t)/dt where lambda(t) is any arbitrary function of t and use it to evaluate path integrals which have not been evaluated previously. We also employ this method to arrive at a formal solution of the propagator for both Levy flights and Brownian subjected to a time-dependent harmonic potential in the underdamped limit of friction. (C) 2015 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
Publisher:	ELSEVIER SCIENCE BV
Additional Information:	Copy right for this article belongs to the ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS
Keywords:	Phase space path integrals; Time-dependent harmonic oscillator; Density matrix; Underdamped limit of friction
Department/Centre:	Division of Chemical Sciences > Inorganic & Physical Chemistry
Date Deposited:	06 Jul 2015 06:40
Last Modified:	06 Jul 2015 06:40
URI:	http://eprints.iisc.ac.in/id/eprint/51803

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