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AIP conference proceedings

Group theory and the harmonic oscillator: The work of Marcos Moshinsky

J. Flores, A. Frank, T. H. Seligman

Published Jan 1, 1992• 0 citations
Publisher
DOI
10.1063/1.42853

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One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1].However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position.Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners.To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schrödinger-like standard equation is applied to the PDMSE.In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD) m(x).Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.

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The group‐theoretical structure of the harmonic oscillator appears in many guises. Originally developed by Marcos Moshinsky among several others for applications in nuclear physics, we point out here that the harmonic oscillator structure appears in aberrations of geometric optics, particularly in their classification by rank, symplectic spin and weight. And further, the finite harmonic oscillator appears again in the nonlinear transformations of finite Hamiltonian systems, when applied to the parallel processing of signals.
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1].However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position.Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners.To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schrödinger-like standard equation is applied to the PDMSE.In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD) m(x).Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.