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Journal of Mathematical Physics

The complete exact solution to the translation-invariant N-body harmonic oscillator problem

Richard L. Hall, Bernd Schwesinger

Published Dec 1, 1979• 22 citations
Publisher

Abstract

It is shown that Schrödinger’s nonrelativistic equation for a translation-invariant system of N particles with arbitrary masses interacting by harmonic pair potentials with arbitrary coupling constants is exactly soluble. An explicit matrix KV is given whose eigenvalues and eigenvectors determine all the exact energies and corresponding eigenfunctions of the N-particle problem. The result is extended to include systems composed of an arbitrary number of groups of identical particles.

DOI
10.1063/1.524038

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Journal of Mathematical Physics•May 1, 1975•61 citations
It is shown that Schrödinger’s nonrelativistic equation for a translation-invariant system of N particles with arbitrary masses interacting by harmonic pair potentials with arbitrary coupling constants is exactly soluble. An explicit matrix KV is given whose eigenvalues and eigenvectors determine all the exact energies and corresponding eigenfunctions of the N-particle problem. The result is extended to include systems composed of an arbitrary number of groups of identical particles.
It is shown that Schrodinger's equation for a translation-invariant system consisting of N particles with arbitrary masses interacting via Hooke's law pair potentials with the same coupling constant can be solved exactly; explicit solutions are found for the case N=3. Exact solutions are also found explicitly for the translation-invariant problem in which a particle with mass m 0 interacts with N identical particles of mass m 1 via a Hooke's law pair potential with coupling constant k 0 2 , and the identical particles interact with each other via Hooke's law pair potentials with coupling constant k 1 2 . The latter solution provides a basis problem for an energy lower-bound method for translation-invariant atom-like systems.
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.
Abstract The resolution of the Schrödinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N − 1 . It has been previously established that the diagonalization can be analytically performed in specific situations, such as for N ≤ 5 or for N identical particles. We show that the matrix J is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.
A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
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.