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Pramana

An iteration algorithm for the time-independent fractional Schrödinger equation with Coulomb potential

Marwan Al‐Raeei, Moustafa Sayem El‐Daher

Published Oct 29, 2020• 10 citations
Publisher
DOI
10.1007/s12043-020-02019-3

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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.
The belief propagation algorithm has been recognized in the information theory community as a soft-decision iterative decoding algorithm. It is the most powerful algorithm found so far for attacking hard optimization problems in channel decoding. Quantum mechanics is the foundation of modern physics with the time-independent Schrodinger equation being one of the most important equations. This paper shows that the equation can be derived from a generalized belief propagation algorithm. Such a connection on a mathematical basis might shed new insights into the foundations of quantum mechanics and quantum computing.