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

Abstract

The class of the even-power series potentials:V(r)=-D+ Σ_k^{\infty} V_kλ^kr^{2k+2}, Vo=ω^2>0, is studied with the aim of obtaining approximate analytic ex­pressions for the energy eigenvalues, the expectation values for the potential and the kinetic energy operator, 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 especially for the treatment of that type 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 and the mass of the particle, and are then applied to the case of the Gaussian potential and to the potential V(r)=-D/cosh^2(r/R). 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.