Exact quantum-mechanical solution for the one-dimensional harmonic oscillator model asymmetrically confined into the infinite well

Abstract

We constructed a new model of the non-relativistic quantum harmonic oscillator within the canonical approach. It is asymmetrically confined into the infinitely high potential well. Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. It is shown that the model has exact solutions. Analytical expression of its wavefunctions of the stationary states is expressed through the Jacobi polynomials , whereas, energy spectrum is non-equidistant. At the limit, when confinement parameters a and b , defining the position of two walls of the potential well go to infinity, then obtained expressions of the wavefunctions also recover the wavefunctions expressed through the Hermite polynomials. • An exactly solvable new model of the non-relativistic quantum harmonic oscillator asymmetrically confined into the infinitely high potential well. • Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. • Energy spectrum is non-equidistant. • The model has correct limit to the so-called Hermite oscillator when position of both walls of the potential well go to infinity.