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University Microfilms International eBooks

The probabilistic solution of the third boundary value problem for the Schrödinger equation and its path integral representation

Vassilis G. Papanicolaou

Published Jan 1, 1988• 0 citations
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Luis A. Rincón

In this paper we prove a path integral representation in configuration space for the solution of a stochastic Schrödinger equation driven by a continuous semimartingale. We follow the approach of path integration established by Itô Albeverio and Høegh-Krohn. The equation we consider is the classical Schrödinger equation plus the stochastic term K(t, x)Ψt(x) o dMt, where Mt is a continuous semimartingale. As an application of our result we solve the equation for a stochastic quantum harmonic oscillator. We solve the path integral representation twice, first by discretising the path integral and also by a change of variable. We finally give the solution to the corresponding stochastic heat equation.

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Imaginary-Time Path Integrals from Klein-Gordon Equation

Gian Fabrizio De Angelis, Maurizio Serva

The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .

Europhysics Letters (EPL)•Mar 15, 1992•3 citations
In this paper we prove a path integral representation in configuration space for the solution of a stochastic Schrödinger equation driven by a continuous semimartingale. We follow the approach of path integration established by Itô Albeverio and Høegh-Krohn. The equation we consider is the classical Schrödinger equation plus the stochastic term K(t, x)Ψt(x) o dMt, where Mt is a continuous semimartingale. As an application of our result we solve the equation for a stochastic quantum harmonic oscillator. We solve the path integral representation twice, first by discretising the path integral and also by a change of variable. We finally give the solution to the corresponding stochastic heat equation.
The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .