WKB quantization of the Morse Hamiltonian and periodic meromorphic functions

Abstract

We show that the well known fact concerning a coincidence between the leading-order WKB and exact quantum mechanical results for an energy spectrum of the Morse Hamiltonian is due to the following property of the bound-state wavefunctions in the complex plane. The logarithmic derivatives of the bound-state eigenfunctions of the Morse Hamiltonian are periodic functions with a pure imaginary period. We show that the Morse potential is the only potential having this property in the following class of potentials: .