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College Physics

Energy spectrum equation of deformed harmonic oscillator potential

Qiu Wei-gang

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Three kinds of deformed harmonic oscillator potentials are given: left and right part with different parameters,left square potential and right harmonic oscillator potential,harmonic oscillator potential plus delta potential.The eigen-state of these potentials can be expressed in terms of the Hermite function.The energy spectrum equations are derived from the matching condition of wave function and its first derivative at origin.

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Three kinds of deformed harmonic oscillator potentials are given: left and right part with different parameters,left square potential and right harmonic oscillator potential,harmonic oscillator potential plus delta potential.The eigen-state of these potentials can be expressed in terms of the Hermite function.The energy spectrum equations are derived from the matching condition of wave function and its first derivative at origin.
We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.
The Schrodinger equation with quantum deformation for three-dimensional Harmonic Oscillator plus Inverse Quadratic potential was obtained the solution by Hypergeometric method with substituting variable and parameter to reduce Schrodinger equation into second order Hypergeometric differential equation.The non-relativistic energy was obtained from condition a=-n and wave function was expressed by Hypergeometric function.The results were calculated and visualized by Matlab R2013a software.The energy spectrum and wave function depend on the quantum deformation and quantum number parameters.