Dresden 2017 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 15: Quantum Dynamics, Decoherence, Quantum Information
DY 15.6: Vortrag
Dienstag, 21. März 2017, 10:45–11:00, ZEU 160
A control theory approach to the Schrödinger equation — •Jeanette Köppe1, Wolfgang Paul1, and Wilfried Grecksch2 — 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 things: 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 non-stationary and the stationary case and establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. Using this method, systems without an exact known wave function, e.g. one-dimensional double-well potential, can be analyzed and an approximation to the wave function can be found.