Hamburg 2016 – wissenschaftliches Programm

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 7: Statistischer Zugang zur Quantentheorie

MP 7.2: Vortrag

Donnerstag, 3. März 2016, 08:50–09:10, VMP6 HS B

Stochastic optimal control, forward-backward stochastic differential equations and the Schrödinger equation — •Wolfgang Paul¹, Jeanette Köppe¹, and Wilfried Grecksch² — ¹Institut für Physik, Martin Luther Universität, 06099 Halle — ²Institut für Mathematik, Martin Luther Universität, 06099 Halle

The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schrödinger equation. We show a way to go around it. This way is based on the derivation of the Schrödinger equation from conservative diffusion processes by E. Nelson [1] and the establishment of (several) stochastic variational principles leading to the Schrödinger equation under the assumption of a kinematics described by Nelsons diffusion processes, in particular by M. Pavon [2].

Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelsons stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schrödinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schrödinger equation and exemplify the approach for a simple 1d problem.

[1] E. Nelson, Phys. Rev. 150, 1079 (1966)

[2] M. Pavon, J. Math. Phys. 36, 6774 (1995)