Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 64, pp. 1-15.
Title: Matrix elements for sum of power-law potentials in
quantum mechanic using generalized hypergeometric functions
Authors: Qutaibeh D. Katatbeh (Jordan Univ., Irbid, Jordan)
Ma'zoozeh E. Abu-Amra (Jordan Univ., Irbid, Jordan)
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
$$
\psi _n(r)= \sqrt{{\frac{2\beta ^{\gamma/2}(\gamma )_n}
{n!\Gamma(\gamma )}}} r^{\gamma - 1/2}
e^{-\frac{\sqrt{\beta }}{2}r^q} \ _pF_1
( -n,a_2,\ldots ,a_p;\gamma;\sqrt {\beta } r^q),
$$
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.