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AIP conference proceedings

Numerical Solution Of The Time-Dependent Schrodinger equation

Bernardine Renaldo Wong, Swee-Ping Chia, Kurunathan Ratnavelu, Muhamad Rasat Muhamad

Published Jan 1, 2009• 2 citations
Publisher

Abstract

We implement a method to numerically solve the Time‐Dependent Schrödinger Equation (TDSE) in one dimension. The time‐dependence is modeled using the Crank‐Nicholson formula. The discrete transparent boundary conditions are employed using a Numerov approximation formula. A normalized Gaussian wavefunction is propagated across a potential step and across a potential well of modified Poschl‐Teller form, respectively. Preliminary results will be presented.

DOI
10.1063/1.3192278

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We implement a method to numerically solve the Time‐Dependent Schrödinger Equation (TDSE) in one dimension. The time‐dependence is modeled using the Crank‐Nicholson formula. The discrete transparent boundary conditions are employed using a Numerov approximation formula. A normalized Gaussian wavefunction is propagated across a potential step and across a potential well of modified Poschl‐Teller form, respectively. Preliminary results will be presented.
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In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.
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