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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 10: Quantum Systems, Symmetries and Scattering

MP 10.1: Vortrag

Donnerstag, 22. März 2018, 14:00–14:20, Z6 - SR 1.012

Derivation of Quantum Hamilton Equations of Motion — •Jeanette Köppe¹, Markus Patzold¹, Michael Beyer¹, Wilfried Grecksch², and Wolfgang Paul¹ — ¹Institut für Physik, MLU Halle-Wittenberg, Germany — ²Institut für Mathematik, MLU Halle-Wittenberg, Germany

Non-relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson’s stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics.

We show that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton’s principle of least action, allows to derive two aspects: i) the Schrödinger equation as the Hamilton-Jacobi-Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton’s equations of motion to the quantum world. We derive their general form for the n-dimensional, non-stationary and the stationary case. The resulting quantum Hamilton equations of motion can be solved (numerically) without knowledge on the wave function and are analyzed for many different systems, e.g. one-dimensional double-well potential or hydrogen atom.