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Journal of Physics A Mathematical and General

Hydrogen atom in the four-dimensional oscillator representation: matrix elements of zn

Hilda A. Cerdeira

Published Oct 1, 1985• 8 citations
Publisher

Abstract

The authors presents a calculation of the matrix elements of z n between four-dimensional harmonic oscillator states. It is shown that a knowledge of the scalar product of two harmonic oscillator states of different frequencies is enough to generate the matrix elements of any power of z. These can be found by taking appropriate derivatives of the aforementioned scalar product.

DOI
10.1088/0305-4470/18/14/022

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•Apr 9, 2016•0 citations
The authors presents a calculation of the matrix elements of z n between four-dimensional harmonic oscillator states. It is shown that a knowledge of the scalar product of two harmonic oscillator states of different frequencies is enough to generate the matrix elements of any power of z. These can be found by taking appropriate derivatives of the aforementioned scalar product.
Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.
Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.