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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

The exact solution of theN-particle schrödinger equation with two-particle harmonic interactions

H. Almström

Published Jun 1, 1969• 4 citations
Publisher
DOI
10.1007/bf02753975

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We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.
Continuation of previous work (ibid., vol.11, no.7, p.1227 (1978)). Earlier exact solutions of Schrodinger's equation for translation-invariant systems of particles interacting by Hooke's law pair potentials are augmented to include systems consisting of an arbitrary number S of groups of identical particles. Exact solutions can always be found whenever the number of distinct masses plus the number of distinct coupling constants does not exceed (2S+1). The case S=2 is solved in detail and is applied to show that harmonic matter is never stable.
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.