Stochastic Methods in Quantum Mechanics

Alladi Ramakrishnan, Nagarajan Ranganathan

Lech S. Papiez

Classical mechanics is formulated as a kind of deterministic optimal control. The simplest, from the optimal control point of view, stochastic generalization of classical mechanisms is submitted. The connections between stochastic mechanics and quantum mechanics are shown, thanks to the stochastic optimal control method.

Kunio Yasue

A time-symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics. The main idea is based on Nelson’s probability theoretical approach to quantum mechanics. His approach is reformulated as a time-symmetric stochastic control problem. Several different control constraints equivalent to Nelson’s are obtained. One of them has a close connection to the Lagrangian formalism of classical mechanics. This suggests to us the use of stochastic calculus of variations. Within the realm of this time-symmetric stochastic control theory it is shown why Schrödinger’s original variational method of quantization was successful. Several advantageous points of the stochastic control theoretical approach to quantum mechanics, including the analysis of the classical limit, are also discussed.

Walter Lehr

Ramabhadra Vasudevan

Masahiko Kanenaga, Mikio Namiki

Introducing the generalized Langevin equation, we extend the stochastic quantization method so as to deal with singular dynamical systems beyond the ordinary territory of quantum mechanics. We also show how the uncertainty relation is built up to the quantum mechanical limit with respect to fictitious time, irrespective of its initial value, within the framework of the usual stochastic quantization method.