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arXiv:2008.03698 · Physics Letters A

On the exact discretization of Schrödinger equation

Chih‐Lung Chou

Published Oct 28, 2020• 2 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2020.126986

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Based on the well-known relation between Fokker-Planck equations and Schrödinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schrödinger equations of “discrete” QM. The latter is a natural discretization of QM and its Schrödinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable “discrete”QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.
Abstract We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of the discrete Schrödinger equation that is used to write the solution for quantum graphs. The formulation of the problem and derivation of secular equation for arbitrary quantum graphs is presented. The application of the approach for the star graph is demonstrated by obtaining eigenfunctions and eigenvalues explicitely. The practical application of the model in conducting polymers and branched molecular chains is discussed.
The importance of discreteness of electron charge for mesoscopic electric circuit is addressed. A quantization of the circuit in accord with the discreteness of electric charges is proposed, and a Schrödinger equation in ‘charge representation’ is obtained. The Schrödinger equation is turned to be Mathieu equation in ‘current representation’, the wavefunction and energy spectrum can be solved exactly. The WKB solutions of the energy levels for the ground state and the excited states are given.