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Proceedings of the Physical Society

Stark states and O(4) symmetry of hydrogenic atoms

J. W. B. Hughes

Published Aug 1, 1967• 91 citations
Publisher

Abstract

The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficient

DOI
10.1088/0370-1328/91/4/306

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The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficient
The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j 1 j 2 l; m 1 m 2 m) with complex j 1 j 2 m 1 m 2 , and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n 2 , the special values, 0 and -1, respectively.
The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates, as a consequence of the ‘‘hidden symmetry’’ stemming from the fixed spatial orientation of the classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the major axis of the elliptic orbit. The separation constant α is not an elementary function of Z or R or quantum numbers. However, for given principal quantum number n and angular momentum projection m, the allowed values of α and corresponding eigenfunctions in spheroidal coordinates are readily obtained from a secular equation of order n−m. We evaluate α(n,m;ZR) and the coefficients gl(α) that specify the spheroidal eigenfunctions as hybrids of the familiar ‖nlm〉 hydrogen-atom states with fixed n and m but different l values. Explicit formulas and plots are given for α and gl and for the probability distributions derived from the hybrid wave functions, ∑lgl(α)‖nlm〉, for all states up through n=4. In the limit R→∞ these hybrids become the solutions in parabolic coordinates, determined simply by geometrical Clebsch–Gordan coefficients that account for conservation of angular momentum and the hidden symmetry. We also briefly discuss some applications of the spheroidal eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for special values of R and the nuclear charge ratio Za/Zb.
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.
Simple and efficient methods for adding angular momenta and for finding angular momentum eigenstates for systems of equivalent electrons are developed. Seniority quantum numbers and coefficients of fractional parentage are presented in a simple manner which does not require 6- and 9-j symbols or knowledge of group theory. Several different common representations are used in specific examples. The transformation matrices between jj- and LS-coupled states are explicitly introduced as an example of change of representation and are used to find matrix elements of the spin-orbit interaction and to examine the Zeeman effect in the two representations. The material is suitable for a graduate course in quantum mechanics.