Matrix elements for the modified P�schl?Teller potential

Abstract

m Analytical exact expressions are obtained for matrix elements of the modified Poschl-Teller oscillator over different operators including powers of the hyperbolic functions sinh( a x ) , cosh(a x ) , and tanh( a x) and the differential operators d / d x and d 2 / d x 2 . These expressions are derived using explicitly the Poschl-Teller eigenfunctions. In addition, several recursion relations connecting different Poschl-Teller matrix elements are obtained using the factorization and hypervirial techniques. It is shown that these relations can be used to make easier the computation of the matrix elements. 0 1996 fohn Wiley & Sons, Inc.Let us consider the evaluation of the matrix elements of powers of sirMax), cosh(ax), and 44 VOL. 57, NO. 1

On an algebraic approach to the Kratzer oscillator

“…It must be stressed that equations (59), ( 60) and (63) differ significantly from equations (34), ( 36) and (10), respectively, in the Karimi and Setare study [19]. It is easy to demonstrate that this fact is a consequence of the improper assumption that γ 2 = 1.…”

“…Construction of coherent states and ladder operators for anharmonic oscillators, described by exactly solvable potentials, is one of the important tasks to realize within the domain of quantum mechanics. The ladder operators can be constructed using the algebraic or supersymmetric approaches [1][2][3][4], which have been successively applied to the Morse, Pöschl-Teller, radial harmonic and other oscillators [5][6][7][8][9][10]. Employing the formalism of the factorization method, Infeld and Hull [11] introduced a new operational procedure, enabling the Schrödinger equation with various solvable potentials to be analytically solved.…”

Matrix elements for powers ofx-dependent operators for the hyperbolic Pöschl–Teller potentials

“…Various matrix elements can be deduced from this closed form which is written in a more compact form than in [9]. For example, we have…”

“…Concerning the bending motions and/or out-of-plane modes, another exactly solvable model, namely the hyperbolic modified Pöschl-Teller (MPT) potential, may be considered and seen as a better alternative than the usual harmonic oscillator. For this potential, matrix elements of the natural variable u = tanh(αx) and of the differential operators d/dx and d 2 /dx 2 have also been derived [9][10][11] with the same techniques as those used for the Morse and Kratzer oscillators. There exist other formulations for Pöschl-Teller-type potentials which are exactly solvable for both the bound and scattering states.…”