Exact solution of position dependent mass Schrödinger equation by supersymmetric quantum mechanics

Abstract The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

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Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

IU Journal of Undergraduate Research ◽

10.14434/iujur.v4i1.24522 ◽

2018 ◽

Vol 4 (1) ◽

pp. 47-55

Author(s):

Timothy Brian Huber

Keyword(s):

Quantum Mechanics ◽

Schrödinger Equation ◽

Harmonic Oscillator ◽

Mechanical System ◽

Schrodinger Equation ◽

Supersymmetric Quantum Mechanics ◽

Time Dependent ◽

Quantum Mechanical ◽

Quantum Mechanical System ◽

Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

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EXACT SOLUTION OF THE SCHRÖDINGER EQUATION FOR THE MODIFIED KRATZER'S MOLECULAR POTENTIAL WITH POSITION-DEPENDENT MASS

International Journal of Modern Physics E ◽

10.1142/s0218301308010428 ◽

2008 ◽

Vol 17 (07) ◽

pp. 1327-1334 ◽

Cited By ~ 22

Author(s):

RAMAZÀN SEVER ◽

CEVDET TEZCAN

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

State Energy ◽

Bound State ◽

Bound State Energy ◽

Energy Eigenvalues ◽

Molecular Potential ◽

Dependent Mass ◽

Morse Potentials ◽

Position Dependent Mass

Exact solutions of Schrödinger equation are obtained for the modified Kratzer and the corrected Morse potentials with the position-dependent effective mass. The bound state energy eigenvalues and the corresponding eigenfunctions are calculated for any angular momentum for target potentials. Various forms of point canonical transformations are applied.

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