Generalized hypervirial and Blanchard's recurrence relations for radial matrix elements

Abstract

Based on the Hamiltonian identity, we propose a generalized expression of the second hypervirial for an arbitrary central potential wavefunction in arbitrary dimensions D. We demonstrate that the new proposed second hypervirial formula is very powerful in deriving the general Blanchard's and Kramers' recurrence relations among the radial matrix elements. As their useful and important applications, we derive all general Blanchard's and Kramers' recurrence relations and some identities for the Coulomb-like potential, harmonic oscillator and Kratzer oscillator. The recurrence relation and identity between the exponential functions and the powers of the radial function are established for the Morse potential. The corresponding general Blanchard's and Kramers' recurrence relations in 2D are also briefly studied.

Bound state solutions ofd-dimensional Schrödinger equation with Eckart potential plus modified deformed Hylleraas potential

“…Such problems have been solved for relativistic and non-relativistic cases with different potentials. [26][27][28][29][30][31][32][33] The rest of the present paper is organized as follows. In Section 2, we briefly review the Nikiforov-Uvarov method in its parametric form.…”

Exact solutions of the radial Schrödinger equation for some physical potentials

“…The second method is related to the supersymmetry (SUSY) method, and, more closesly, with the factorization method [8]. The exact quantization rule (EQR) has also shown its power in calculating the energy levels of all the bound states for some exactly solvable quantum systems such as the Morse, the Rosen-Morse, the Kratzer, the harmonic oscillator, the first and second Pöschl-Teller and the pseudoharmonic oscillator potentials [9][10][11][12]. The Nikiforov-Uvarov (NU) method [13,14] has been introduced for the solution of the Schrödinger equation to its energy levels by transforming it into hypergeometric type second order differential equations.…”

A General Approach for the Exact Solution of the Schrödinger Equation

“…In some cases, one may need a second solution of (10). In this case, if the same procedure is followed by using…”