J. Morales

Abstract The solution to a spectral problem involving the Schrödinger equation for a particular class of multiparameter exponential‐type potentials is presented. The proposal is based on the canonical transformation method applied to a general second‐order differential equation, multiplied by a function g ( x ), to convert it into a Schrödinger‐like equation. The treatment of multiparameter exponential‐type potentials comes from the application of the transformed results to the hypergeometric equation under the assumption of a specific g ( x ). Besides presenting the explicit solutions and their spectral values, it is shown that the problem considered in this article unifies and generalizes several former studies. That is, the proposed exactly solvable multiparameter exponential‐type potential can be straightforwardly applied to particular exponential potentials depending on the choice of the involved parameters as exemplified for the Hulthén potential and their isospectral partner. Moreover, depending on the function g ( x ), the proposal can be extended to find different exactly solvable potentials as well as to generate new potentials that could be useful in quantum chemical calculations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Abstract A second‐quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one‐dimensional harmonic oscillator matrix elements. The most general case of 〈 m | f (â, â + )| n 〉 is considered, and the recurrence relations for f (â, â) = X k , exp(−β X ), and exp(− X 2 ) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum‐rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two‐center integrals.

As is well known, the binomial theorem is a classical mathematical relation that can be straightforwardly proved by induction or through a Taylor expansion, albeit it remains valid as long as [A,B]=0. In order to generalize such an important equation to cases where [A,B]≠0, an algebraic approach based on Cauchy’s integral theorem in conjunction with the Baker–Campbell–Hausdorff series is presented that allows a partial extension of the binomial theorem when the commutator [A,B]=c, where c is a constant. Some useful applications of the new proposed generalized binomial formula, such as energy eigenvalues and matrix elements of power, exponential, Gaussian, and arbitrary f(x̂) functions in the one-dimensional harmonic oscillator representation are given. The results here obtained prove to be consistent in comparison to other analytical methods.