Solution of the Schrödinger Equation in Harmonic Oscillator Representation

1988 ◽  
pp. 156-162
Author(s):  
Erich W. Schmid ◽  
Gerhard Spitz ◽  
Wolfgang Lösch
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Oscillator Representation

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

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.

scholarly journals Discrete Quantum Harmonic Oscillator

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1362
Author(s):  
Alina Dobrogowska ◽  
David J. Fernández C.
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Discrete Model ◽  
Schrodinger Equation ◽  
Factorization Method ◽  
Quantum Harmonic Oscillator ◽  
Meixner Polynomials

In this paper, we propose a discrete model for the quantum harmonic oscillator. The eigenfunctions and eigenvalues for the corresponding Schrödinger equation are obtained through the factorization method. It is shown that this problem is also connected with the equation for Meixner polynomials.

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