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Computers & Mathematics with Applications

Closed-form solutions of the Schrödinger equation for a coupled harmonic potential in three dimensions

Ram Mehar Singh, S.B. Bhardwaj, Shubham Mishra

Published Jun 29, 2013• 10 citations
Publisher
DOI
10.1016/j.camwa.2013.04.030

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We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz Ψ(r)=rae−λr2ψ(r)Ψ(r)=r^{a}e^{-λr^2} ψ(r)Ψ(r)=rae−λr2ψ(r) we transform the time independent Schrödinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in one-dimension harmonic oscillator and interacting through finite distance soft-core potential. In such special case the eigen value equation become the Weber's differential equation and its solution are also given in the form of confluent hypergeometric function.