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International Journal of Quantum Chemistry

A separable model of N interacting Particles

Jacek Karwowski

Published Jan 1, 2008• 10 citations
Publisher

Abstract

Abstract A separable model of N particles interacting by two‐particle potentials is presented. The interactions between disjoint pairs of particles are described by arbitrary two‐particle potentials which depend on the interparticle distance only. The remaining interactions are described by two‐particle harmonic oscillator potentials. A set of the pair potentials for which the model is quasi‐exactly solvable is derived. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

DOI
10.1002/qua.21751

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Abstract A separable model of N particles interacting by two‐particle potentials is presented. The interactions between disjoint pairs of particles are described by arbitrary two‐particle potentials which depend on the interparticle distance only. The remaining interactions are described by two‐particle harmonic oscillator potentials. A set of the pair potentials for which the model is quasi‐exactly solvable is derived. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
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.
We have obtained an exact analytical solution for a system of two particles moving under the action of a central harmonic-oscillator potential and the mutual Coulomb interaction augmented by a potential linear in the interparticle coordinate. The expressions for the single-particle density, the exchange-correlation energy quantities, and the correlation functions are thereby derived. Numerical results on the exchange-correlation contributions are presented using plots of various local energy densities as well as the corresponding global energy quantities for varying strengths of the oscillator potential.
Abstract The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr2, with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
In this paper we will report on a one-dimensional, non-separable quantum many-particle system.It consists of two (distinguishable) particles moving on the half-line R + being subjected to two different kinds of two-particle interactions: singular many-particle interactions localized at the origin and a binding-potential leading to a molecular-like state.We will formulate the model precisely, obtaining a well-defined self-adjoint operator (the Hamiltonian for our system) and elaborate on its spectral properties.In addition, we will present possible directions for future research.
Abstract The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011