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Journal of the Korean Mathematical Society

AN APPROACH TO SOLUTION OF THE SCHRÖDINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

Seung Jun Chang, Jae Gil Choi, Hyun Soo Chung

Published Mar 1, 2013• 10 citations
Publisher

Abstract

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 $\ddot{o}$ dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential. The Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

DOI
10.4134/jkms.2013.50.2.259

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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.
In the setting of Cameron and Storvick's recent theory we show that the solution of an integral equation formally equivalent to the Schroedinger equation is expressible as the analytic Feynman integral of a function on ^-dimensional Wiener space of the form F(X) = exp{/ o '0(ίs, X(s) + I) ds}ψ(X(t\+ ?).Here Xis an Revalued continuous function on [0, /] such that Z(0) = 0, ?G R\ and ψ and θ(s, ) are Fourier-Stieltjes transforms.REMARK 3.2.Corollary 3.2 and Lemma 3.2 will yield the measurability properties that we will need as we continue.For example, let θ be in % and for λ > 0 and fin W let G: [0, ί 0 ] X C[0, t 0 ] -> C be defined by The Pacific
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.