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College Physics

Bound state solutions of the one-dimensional Schrdinger equation for the harmonic oscillator via the Laplace transform

Zidong Chen

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

AbstrcatThe second-order one-dimensional Schrdinger differential equation for the harmonic oscillator is reduced to a first-order differential equation by means of the Laplace transform,and then,its solutions are obtained by means of directly making use of an integral.The energy spectrum is derived from the condition that the wave functions must be single-valued.The required eigenfunctions are obtained by applying the expansions of series and inverse Laplace transform.

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•Mar 12, 2023•1 citation
AbstrcatThe second-order one-dimensional Schrdinger differential equation for the harmonic oscillator is reduced to a first-order differential equation by means of the Laplace transform,and then,its solutions are obtained by means of directly making use of an integral.The energy spectrum is derived from the condition that the wave functions must be single-valued.The required eigenfunctions are obtained by applying the expansions of series and inverse Laplace transform.
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Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.