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Journal of Physics A Mathematical Nuclear and General

O(3,1) symmetry of the hydrogen atom

S. Datta Majumdar, D Basu

Published May 1, 1974• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/7/7/004

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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 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
An expansion formula for Stark effect transition probabilities is derived in terms of field-free (spherical) transition probabilities. Expansion coefficients are squares of Clebsch-Gordan coefficients from the SO(4) = SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) description of the hydrogen atom. The method provides new sum rules for Stark transitions which explain regularities in previous calculations. Accurate formulas for average Stark transition probabilities obtain from application of the method to an empirical model for field-free transition probabilities. The approximation permits estimation of Stark lifetimes in the large-nnn regime.
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.
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.