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Journal of Physics A Mathematical and Theoretical

Path integral based transition probability evaluation in a quantum system with a diagonal density matrix of the final state

A B Bichkov, Valery V Smirnov

Published Dec 3, 2012• 8 citations
Publisher

Abstract

The variant of transition probability evaluation in a quantum system, based on transition probability representation by a path integral using a density matrix formalism, is described. To estimate an integral's value, a set of variables is selected and the Monte Carlo method is used for integration over these variables. The stationary phase method is applied to integration over the remaining variables. In the case of the final state with a diagonal density matrix, this scheme leads to a simple expression for transition probability.

DOI
10.1088/1751-8113/46/1/015303

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The variant of transition probability evaluation in a quantum system, based on transition probability representation by a path integral using a density matrix formalism, is described. To estimate an integral's value, a set of variables is selected and the Monte Carlo method is used for integration over these variables. The stationary phase method is applied to integration over the remaining variables. In the case of the final state with a diagonal density matrix, this scheme leads to a simple expression for transition probability.
In present work are considered the scheme of evaluation the transition probability in quantum system. It is based on path integral representation of transition probability amplitude and its evaluation by means of a saddle point method, applied to the part of integration variables. The whole integration process is reduced to initial value problem solutions of Hamilton equations with a random initial phase point. The scheme is related to the semiclassical initial value representation approaches using great number of trajectories. In contrast to them from total set of generated phase paths only one path for each initial coordinate value is selected in Monte Karlo process.
An alternative formulation of the discrete real time coherent state path integral is examined. Following the development of a previous work [T. L. Marchioro II, J. Math Phys. 31, 2935 (1990)] the discrete coherent state action is transformed to second order form. The resulting path integral expression contains an integrand which factors into a Gaussian probability density and an oscillatory term, making the integral directly amenable to Monte Carlo techniques. Several simple examples in one and two spatial dimensions are presented and the possibilities of extending such an approach to general multidimensional systems are discussed.
In this article we present two phase space path integrals in terms of coherent states. The first one is derived in a standard fashion but using a nonstandard resolution of the identity in terms of coherent states with different width parameters. The second path integral emerges from a novel phase space representation in terms of coherent states distributed on n-dimensional manifolds embedded in the 2n-dimensional phase space of an n-degree-of-freedom system. These states are shown to form locally complete basis sets since we show that fairly smooth and localized functions can be expanded in terms of them in a unique way. In this representation the time evolution operator can be cast in the form of a phase space path integral. Both path integrals can be evaluated by straightforward implementation of Monte Carlo methods. In both cases the probability amplitude between two phase points turns out to be proportional to the average of the phase, eih∫(p dq−H dt), over a Markov process of phase space paths consisting of classical trajectories interrupted at intervals by Gaussian ‘‘quantum jumps.’’ The numerical evaluation of these expressions through importance sampling is demonstrated.
Using the discretized path integral formalism we develop a numerically convenient and accurate method for evaluating the finite time propagator (and density matrix) of a given system. The eigenfunctions and energies of large number of states of the system are also found. We demonstrate the accuracy of the method by computing eigenvalues, eigenfunctions, position and flux autocorrelation functions and complex time propagators for a large number of systems and comparing these results with those available from other sources. In all the cases we find very good agreement. The method thus provides a very simple and accurate procedure for the study of static and dynamic properties of a quantum system.