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International Journal of Quantum Chemistry

The system of harmonically interacting particles: An exact solution of the quantum‐mechanical problem

M. S. Osadchii, V. V. Murakhtanov

Published Feb 1, 1991• 14 citations
Publisher

Abstract

Abstract An original procedure has been developed to determine the eigenvalues and the eigenfunctions of quantum‐mechanical many‐body systems with a harmonic oscillator interaction. Exact solutions in the explicit form have been obtained for the case of systems containing three groups of identical particles. For the system containing two groups of identical particles, a solution in the adiabatic approximation has been obtained and compared with the exact one.

DOI
10.1002/qua.560390207

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Abstract An original procedure has been developed to determine the eigenvalues and the eigenfunctions of quantum‐mechanical many‐body systems with a harmonic oscillator interaction. Exact solutions in the explicit form have been obtained for the case of systems containing three groups of identical particles. For the system containing two groups of identical particles, a solution in the adiabatic approximation has been obtained and compared with the exact one.
An exact solution is obtained for a model three-body system in which the particles interact through pairwise, harmonic oscillator, and spin-dependent potentials, the solution being achieved by a simple transformation to an appropriate coordinate system. In particular, it is shown that the S-state wave functions of the ground state of the triton span the two-dimensional representation of the symmetric group S3, are orthogonal, and resemble closely the Irving wave forms found so successful in variational calculations.
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 describe a procedure to solve an up to problem where the particles are separated topologically in N groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator potentials. The problem is first reduced to an analytically solvable N -body problem and N independent two-body problems. We calculate analytically spectra, wave functions, and normal modes for both the inverse square and delta-function two-body interactions. In particular, we calculate separation energies between two strings of particles. We find that the string separation energy increases with N and interaction strength.
The adiabatic approximation is set up to estimate variationally the energy of a quantum particle interacting with a quantum harmonic oscillator. This example is seen as a prototype for a wide range of problems, including molecular clusters, free or in condensed matter. A heirarchy of approximations is introduced, labelled simplified adiabatic, average field, and classical field. The quasiadiabatic limit is then introduced, in which the oscillator takes account, as much as possible, of the instantaneous particle position, and an interpolation procedure between the average field adiabatic and quasiadiabatic limits is derived. Questions of implementation are discussed. In particular, an approximation is introduced, in which the quasiadiabatic generalization of the average field case is found to be energetically favored, and which reduces to a problem similar to the static-field case, for which many-electron computational methods are well developed. Now, however, oscillator zero-point effects are incorporated in a renormalized oscillator self-energy and a modified effective particle-oscillator interaction.