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Four-body quantum system in the presence of solvable potential

Zahra Bakhshi, Sare Khoshdooni

Published Feb 17, 2020• 0 citations
Publisher

Abstract

In this article, a four-particle quantum system is introduced with energy discrete spectrum including a harmonic potential and a three-body interaction potential. By defining the Jacobi coordinates for each particle, separately, one coordinate is eliminated as a transition in the energy spectrum. Then the system is studied in polar coordinates and by using the variables separation method, the schrödinger equation of the system is transformed into three separate differential equations. Therefore, energy eigenvalues and wave eigenfunctions are calculated in each dimension. Also, The wave eigenfunctions figures are investigated in one and three dimensions.

DOI
10.22541/au.158199810.01227813

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We consider a class of system of N interacting particles in any dimension—the potential includes a quadratic pair potential and an arbitrary translation-invariant position-dependent potential that is homogeneous of degree −2. The group SU(1,2)¯ (1̄,1̄) is the dynamical group for the Hamiltonian. We illustrate the significance of the Casimir operator in relation to the separation of variables method; obtain a series of eigenfunctions that transform under the unitary irreducible representations of SU(1,2)¯ (1̄,1̄) labeled by the ground state energy; indicate the structure of arbitrary eigenfunctions; and specify when the complete energy spectrum is linear. We treat N-body examples which include two- and three-body forces. For N identical particles in one dimension interacting with a quadratic pair potential and an inverse square pair potential, we exhibit a series of eigenfunctions characterized by four quantum numbers. These eigenfunctions reduce to the complete set of eigenfunctions for five particles. We indicate how a complete set of eigenfunctions for N particles are obtained.
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.
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Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.
This chapter presents a detailed examination of a number of relatively simple problems pertaining to the scattering and bound states of two structureless particles interacting through a central field. It describes the spectrum of the angular momentum operators by a method that does not refer at all to the structure of the system. The spherical harmonics are the eigenfunctions appropriate to a single particle that can be described by the three classical degrees of freedom, or for a pair of such particles in their center-of-mass frame. The formulation of collision theory that is based on the one-body Schrodinger equation with a Hermitian interaction. The chapter introduces and discusses the observables pertaining to the two-body problem and explores the eigenvalues and eigenfunctions of these observables. Since the degeneracies of the energy spectrum are a consequence of the symmetries of the Hamiltonian, they can be determined experimentally by destroying this symmetry.