Symmetry (Mar 2024)

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

  • Anna Mazhar,
  • Jeremy Canfield,
  • Wesley N. Mathews,
  • James K. Freericks

DOI
https://doi.org/10.3390/sym16030297
Journal volume & issue
Vol. 16, no. 3
p. 297

Abstract

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Using a flexible form for ladder operators that incorporates confluent hypergeometric functions, we show how one can determine all of the discrete energy eigenvalues and eigenvectors of the time-independent Schrödinger equation via a single factorization step and the satisfaction of boundary (or normalizability) conditions. This approach determines the bound states of all exactly solvable problems whose wavefunctions can be expressed in terms of confluent hypergeometric functions. It is an alternative that shares aspects of the conventional differential equation approach and Schrödinger’s factorization method, but is different from both. We also explain how this approach relates to Natanzon’s treatment of the same problem and illustrate how to numerically determine nontrivial potentials that can be solved this way.

Keywords