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arXiv:quant-ph/0003081 · Physics Letters A

The Coulomb – harmonic oscillator correspondence in symmetric quantum mechanics

Miloslav Znojil, GÉZA LÉVAI

Published Jul 1, 2000• 64 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(00)00400-x

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We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential V(x)=−γ/∣x∣V(x)=-γ/|x|V(x)=−γ/∣x∣. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.
The function exp(iwt) describes an oscillating motion. Energy of the oscillator is proportional to the square of w. exp(iwt) is the solution of a differential equation. We have replaced this differential equation by the corresponding finite-time difference equation with d as the smallest span of time. We have, then, symmetrized the equation so that it remains invariant under the change d going to -d. This symmetric equation has solutions in pairs. The angular speed w is modified to w' or w". w' contains a part with an integer. w" contains a part with a half-integer. This corresponds to quantum mechanical oscillator energy levels. F= a.exp(iwt) describes oscillation between -a and +a. If we make w=0, F describes free oscillation between -a and +a. Reciprocal symmetric oscillator, thus, unifies quantum and classical harmonic oscillators on one hand, and a harmonic oscillator and a free particle in a potential well on the other hand. No quantum mechanical postulates are involved.