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arXiv:math-ph/0512079 · Journal of Physics A Mathematical and General

Exact solutions for semirelativistic problems with non-local potentials

Richard L Hall

Published Jan 11, 2006• 6 citations
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Abstract

It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form , where is a non-local potential with a separable kernel of the form . Explicit examples in one and three dimensions are discussed, including the Yamaguchi and Gauss potentials. The results are used to obtain lower bounds for the energy of the corresponding N -boson problem, with upper bounds provided by the use of a Gaussian trial function.

DOI
10.1088/0305-4470/39/4/011

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It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form , where is a non-local potential with a separable kernel of the form . Explicit examples in one and three dimensions are discussed, including the Yamaguchi and Gauss potentials. The results are used to obtain lower bounds for the energy of the corresponding N -boson problem, with upper bounds provided by the use of a Gaussian trial function.
We consider a system of N identical bosons which interact in one dimension via attractive pair potentials and obey nonrelativistic quantum mechanics. This system is studied from two complementary viewpoints. The equivalent two-body method approximates the system as a collection of (N-1) independent two-particle systems with coupling constants enhanced by the factor N/2, and this yields energy lower bounds. The collective-field method approaches the problem from the standpoint of the limit as N\ensuremath{\rightarrow}\ensuremath{\infty} and leads to energy upper bounds. We find that Gaussian trial functions for the N-particle Hamiltonian and Gaussian trial densities in the collective-field theory lead to precisely the same energy upper bounds. The upper bounds provided by field theory allow for systematic improvement via a variational principle. Details are worked out for two exactly soluble problems, namely, the harmonic oscillator and the attractive \ensuremath{\delta} potential.
We study a bound system of N identical bosons interacting by model pair potentials of the form $V(r)=\ensuremath{\lambda}\mathrm{sgn}{(p)r}^{p}+\ensuremath{\mu}{/r}^{2},$ $\ensuremath{\lambda}>0,$ $\ensuremath{\mu}>~0.$ By using a variational trial function and the ``equivalent two-body method,'' we find explicit upper and lower bound formulas for the N-particle ground-state energy in arbitrary spatial dimensions d> 3d>~3d> 3 for the two cases p=2p=2p=2 and $p=\ensuremath{-}1.$ It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory.
We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.
In this paper, we present exact solutions of Schrödinger equation for a class of non-central physical potentials within the formalism of position-dependent effective mass. The energy eigenvalues and eigenfunctions of the bound-states for the Schrödinger equation are obtained analytically by means of asymptotic iteration method and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates. Our results are in excellent agreement with other works in the literature.
In this paper we present exact solutions of Schrodinger equation (SE) for a class of non central physical potentials within the formalism of position-dependent effective mass. The energy eigenvalues and eigenfunctions of the bound-states for the Schrodinger equation are obtained analytically by means of asymptotic iteration method (AIM) and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates. Our results are in excellent agreement with other works in the literature.