A solvable double well

TL;DR

This paper presents an exactly solvable quantum double well model created by joining two shifted harmonic oscillators, analyzes its energy spectrum, and explores time oscillations and modifications with delta potentials.

Contribution

It introduces a novel exactly solvable double well potential by combining harmonic oscillators and extends the analysis to include delta-function perturbations.

Findings

Exact energy eigenvalues derived from wavefunction continuity conditions

Oscillations between ground states observed in time evolution

Modified spectrum when delta-function potentials are added

Abstract

We study the quantum behaviour of a particle moving in a one-dimensional double well potential. This double well is obtained by gluing together, at the origin, two shifted harmonic oscillator potentials. The Schr\"odinger equation is exactly solvable. The requirement that discontinuities, in the wavefunction and its first derivative, are absent at the origin, leads to the quantisation of the energy eigenvalues. We also show that oscillations in time take place between two nearby single harmonic oscillator ground states. Finally, the double well potential is augmented by a Dirac delta-function potentials at the origin and the corresponding Schr\"odinger equation is solved.