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Journal of Henan University

Solution of Wave Function and the Eigen-value of Energy for the One-Dimensional Linear Harmonic Oscillator via the Fourier Transform Ation

YU Feng-jun

Published Jan 1, 2008• 0 citations
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Abstract

By the Fourier transform ation,we change the Schr dinger equation of one-dimensional linear harmonic oscillator to the first-order differential equation which can be easily solved.The progressional solution of the Schr dinger equation can be obtained by the inverse Fourier transform ation after we solve the first-order differential equation.Finally,we can fix the eigen-value and eigen-function of energy based on the boundary condition that wave function is zero when the variable becomes infinite.

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By the Fourier transform ation,we change the Schr dinger equation of one-dimensional linear harmonic oscillator to the first-order differential equation which can be easily solved.The progressional solution of the Schr dinger equation can be obtained by the inverse Fourier transform ation after we solve the first-order differential equation.Finally,we can fix the eigen-value and eigen-function of energy based on the boundary condition that wave function is zero when the variable becomes infinite.
The time-dependent Schrdinger equation for the harmonic oscillator with time-dependent frequency ω=ω(t) and mass m=m(0)ω(0)ω(t)exp[2c∫t0ω(t)dt] can be solved by means of simple transformation and separation of variables.The wave functions are obtained for c21,c2=1 and c21,respectively.When |x|→∞,the wave function for c21 tends to zero but it is not so for c2≥1.Finally,some special cases are given for the time-dependent harmonic oscillator and one of them is similar to a damped harmonic oscillator.
In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ⋄{\diamond}⋄ -convolutions. Further, we give an approach to solution of the Schr o¨\ddot{o}o¨ dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr o¨\ddot{o}o¨ dinger equations for harmonic oscillator and double-well potential. The Schr o¨\ddot{o}o¨ dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick,the rigorous solutions of stationary Schro¨dinger equation for one-dimensional harmonic oscillator are solved with the methods of matrix mechanics and wave mechanics respectively.It is showed that the matrix mechanics and the wave mechanics are equivalent in phase space.The eigenfunctions in position and momentum spaces can be obtained through the Fourier-like projection transformation from the phase-space eigenfunction.