A local solution of the Schrödinger equation

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Abstract
A local method is developed for solving the Schrödinger equation. The method is local in the sense that it can determine the value of the solution of the Schrödinger equation at an arbitrary point directly rather than extracting this value from the field solution. The method is based on properties of diffusion processes, the Itô formula, and Monte Carlo simulation. Simplicity, accuracy, and generality are the main features of the proposed local solution. The extension of the proposed method to solve the stochastic version of the Schrödinger equation is elementary. Two examples with Dirichlet and Neumann boundary conditions are presented to demonstrate the application and evaluate the accuracy of the proposed local solution.

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