Application of the quantum mechanical hypervirial theorems to even-power series potentials

Abstract

The class of the even-power series potentials, V(r) = -D + E~=0 Vkkkr 2k+2, V0 = to 2 > 0, is studied with the aim of obtaining approximate analytic expressions for the nonrelativistic energy eigenvalues, the expectation values for the potential and kinetic energy operators, and the mean square radii of the orbits of a particle in its ground and excited states. We use the hypervirial theorems (HVT) in conjunction with the Hellmann-Feynman theorem (HFT), which provide a very powerful scheme for the treatment of the above and other types of potentials, as previous studies have shown. The formalism is reviewed and the expressions of the above-mentioned quantities are subsequently given in a convenient way in terms of the potential parameters, the mass of the particle, and the corresponding quantum numbers, and are then applied to the case of the Gaussian potential and to the potential V(r) = -Dlcosh2(rlR). These expressions are given in the form of series expansions, the first terms of which yield, in quite a number of cases, values of very satisfactory accuracy. ~TRODUCTIONVarious approaches can be employed in an effort to obtain approximate expressions for the energy eigenvalues and other quantities of interest for a particle moving in a certain potential. Unfortunately, the majority entail cumbersome calculations of matrix elements or the ingenious deduction of a trial function. These obstacles can be overcome by using the hypervirial theorem (Hirschfelder, 1960) in conjunction with the Hellmann-Feynman theorem, the HVT-HFT method, which could have been labeled as classical, had it not been for the peculiarities of the potential we select each time.

Quantum information entropies for the $$\ell $$ ℓ -state Pöschl–Teller-type potential

“…This potential is a non-homogeneous potential, as described by Sen and Katriel (2006) [23]. It belongs to a class of even-power series potentials, this class of potentials behave like a harmonic oscillator potential (near the origin), often termed 'oscillator-like' potentials [20,21,25].…”

Two approximate formulae for the binding energies in Λ-hypernuclei and light nuclei

“…can be used as approximations to the self-consistent potential in the study of the behavior of a nucleon in light nuclei [9] and have also been used extensively in hypernuclear physics [10][11][12][13][14][15][16].…”

“…By means of the hypervirial theorems (HVT), a method used extensively in theoretical physics [14][15][16][17][18][19][20][21][22][23][24][25], approximate analytic expressions for the eigenvalues of the potentials in question in the form of s-series were obtained. For instance, for the RPT potential the eigenvalues are…”