Annals of Physics

Solution of the Schrödinger equation in terms of classical paths

Published May 1, 1974• 1 citation

Publisher

DOI: 10.1016/0003-4916(74)90327-3

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Annals of Physics

Solution of the Schrödinger equation in terms of classical paths

Published May 1, 1974• 1 citation

Publisher

DOI: 10.1016/0003-4916(74)90327-3

Related papers

Solution of the Schrödinger equation in terms of classical paths

Roger Balian, Claude Bloch

Annals of PhysicsJun 1, 1974380 citations

An approach to the solution of Schrödinger's equation

Boris V. Alekseev, A. I. Аbakumov

Soviet physics. DokladyFeb 1, 19820 citations

Path integral representation for the solution of a stochastic Schrodinger equation driven by a semimartingale

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.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.

Journal of Interdisciplinary MathematicsJan 1, 20045 citations

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Theo Johnson-Freyd

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on ${\mathbb{R}^n}$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in ${\hbar}$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the ${t \to 0}$ limit approaches the δ distribution. As such, our construction gives explicitly the full ${\hbar\to 0}$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on $${\mathbb{R}^n}$$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in $${\hbar}$$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the $${t \to 0}$$ limit approaches the δ distribution. As such, our construction gives explicitly the full $${\hbar\to 0}$$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

arXiv:1003.1156Letters in Mathematical PhysicsSep 29, 20108 citations

Classical trajectories as solutions of the Schrödinger equation

M.L. Strekalov

Journal of Mathematical ChemistryOct 17, 20150 citations

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on

{\mathbb{R}^n}

, with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in

{\hbar}

, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the

{t \to 0}

limit approaches the δ distribution. As such, our construction gives explicitly the full

{\hbar\to 0}

asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.