An exactly soluble Schrödinger equation with a bistable potential

1980/04/01 by M. Razavy · 175 citations
Mathematics · Physics and Astronomy · #Bistability #Classical mechanics #Diatomic molecule #Eigenfunction #Eigenvalues and eigenvectors #Homonuclear molecule #Mathematical analysis #Mathematical physics #Mathematics #Molecule #Physics #Quantum Mechanics and Applications #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Schrödinger equation #Trigonometric functions #Wave function

paper · doi:10.1119/1.12141

published in American Journal of Physics 48(4), 285-288 (American Institute of Physics)

openalex publication_date 1980/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

For a bistable potential which is the sum of two hyperbolic cosine functions, the Schrödinger equation for the low-lying states of a homonuclear diatomic molecule can be solved analytically. In this model the potential and the energy eigenvalues depend on three parameters, and the resulting wave functions are continuous and have continuous derivatives everywhere. This last property of the wave functions enables one to generate a family of soluble bistable potentials by applying the theorem of Darboux to the discrete eigenfunctions of the Schrödinger equation.

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