Electron. J. Diff. Eqns., Vol. 2008(2008), No. 64, pp. 1-15.

Matrix elements for sum of power-law potentials in quantum mechanic using generalized hypergeometric functions

Qutaibeh D. Katatbeh, Ma'zoozeh E. Abu-Amra

Abstract:
In this paper we derive close form for the matrix elements for $\hat H=-\Delta +V$, where $V$ is a pure power-law potential. We use trial functions of the form
$$ \psi _{n}(r)= \sqrt{{\frac{2\beta ^{\gamma/2}(\gamma )_{n}} {n!\Gamma(\gamma )}}} r^{\gamma - 1/2} e^{-\frac{\sqrt{\beta }}{2}r^q} \ _{p}F_{1} ( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q), $$
for $\beta, q,\gamma >0$ to obtain the matrix elements for $\hat H$. These formulas are then optimized with respect to variational parameters $\beta ,q$ and $\gamma $ to obtain accurate upper bounds for the given nonsolvable eigenvalue problem in quantum mechanics. Moreover, we write the matrix elements in terms of the generalized hypergeomtric functions. These results are generalization of those found earlier in [2], [8-16] for power-law potentials. Applications and comparisons with earlier work are presented.

Submitted February 19, 2008 Published April 28, 2008.
Math Subject Classifications: 34L15, 34L16, 81Q10, 35P15.
Key Words: Schrodinger equation; variational technique; eigenvalues; upper bounds; analytical computations.

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Qutaibeh D. Katatbeh
Department of Mathematics and Statistics, Faculty of Science and Arts
Jordan University of Science and Technology
Irbid 22110, Jordan
email: qutaibeh@yahoo.com
Ma'zoozeh E. Abu-Amra
Department of Mathematics and Statistics, Faculty of Science and Arts
Jordan University of Science and Technology
Irbid 22110, Jordan

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