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Journal of Mathematical Physics

Probabilistic representation formulas for the time evolution of quantum systemsa)

Pedro Pereyra, Roberto B. Quezada

Published Jan 1, 1993• 7 citations
Publisher

Abstract

By using a new procedure representation formulas are obtained for the solution of the Cauchy problem for hyperbolic systems generalizing Schrödinger and Dirac equations. This procedure makes explicit the relation of the underlying Poisson jump process with the perturbation series. It allows one to find out a path integral representation formula for the two-dimensional Dirac equation. Further, it can be applied in the four-dimensional case with a Coulombian potential. This approach is also used to describe the time evolution of dissipative two-level systems and estimate transition probabilities.

DOI
10.1063/1.530387

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By using a new procedure representation formulas are obtained for the solution of the Cauchy problem for hyperbolic systems generalizing Schrödinger and Dirac equations. This procedure makes explicit the relation of the underlying Poisson jump process with the perturbation series. It allows one to find out a path integral representation formula for the two-dimensional Dirac equation. Further, it can be applied in the four-dimensional case with a Coulombian potential. This approach is also used to describe the time evolution of dissipative two-level systems and estimate transition probabilities.
We show that the Liouville equation for any Hamiltonian system can be identified with the Schrödinger equation obtained by the Schrödinger-Dirac quantization of some “extended” Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent some solutions of the Cauchy problem for the Liouville equation by Feynman path integrals [ 1 ] over trajectories in the phase space of the extended Hamiltonian systems.
A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a ccc-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
Rigorous path integral formulas are given which represent, in two space-time dimensions, the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagators for the Dirac particle. It is also shown that the theory can be applied to a free particle and a particle in a central electric field in four space-time dimensions and reveals some aspects of the path integral. Heuristically discussed is the connection with the phase space path integral or Hamiltonian path integral.