A systematic approach for obtaining the Green functions of time-dependent Schrödinger equations by Fourier transform

Gin Yih Tsaur, Jyhpyng Wang

Research output: Contribution to journalArticlepeer-review

Abstract

The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrö dinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.

Original languageEnglish
Article number045402
JournalEuropean Journal of Physics
Volume37
Issue number4
DOIs
StatePublished - 10 May 2016

Keywords

  • Fourier transform
  • Green function
  • time-dependent Schrodinger equation

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