The Schrödinger Equation in Potential Representation

Yifei Zheng, William M. McEneaney, Hidehiro Kaise

A diffusion-based representation for the Maslov dequantization of the time-dependent Schrödinger equation is obtained, where the wave function corresponds to the statical value function of a complex-valued stochastic control problem. Whereas previous efforts along this line were restricted to holomorphic potentials, we here use an additional controller specifically constructed to generate the Coulomb potential via staticization. The solution to the Schrödinger equation is obtained from requantization of the representations. A demonstrative numerical example is included at the end.

Xiaozheng Ma, Costas M. Soukoulis

William M. McEneaney, Peter M. Dower

A stochastic control representation for the Maslov dequantization of the solution of the Schrödinger equation is obtained. The representation employs complex-valued diffusion process and stationarity of an action functional. The use of stationarity removes the need for convexity of the action, and hence there is no restriction on the duration of the problem. Previous efforts along this line were restricted to potentials which had holomorphic extensions to the entire space. The use of a stationarity based representation for the Coulomb potential removes that restriction. The result is a representation in terms of iterated staticization, which is expected to yield exceptional computational benefits.

Paul Bracken, James Hayes

We write the Bohm–Landau wave function in terms of simple hyperbolic numbers and apply this representation to the Schrödinger and Klein–Gordon equations. It is shown that the Schrödinger equation can be separated in one space dimension. The radial part of the wave function and the derivatives of the radial function can be related to the potential and its derivative. We also apply this representation to the Klein–Gordon equation.

JOHN C. MORRISON