Some studies of the theory and application of continuous groups in atomic spectroscopy
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis is concerned with the representation theory of continuous groups both compact and non-compact and its application to atomic spectroscopy. In Chapter I some atomic wavefunctions for equivalent electrons in the group scheme SU2 x (U2l+1 R2l+1 R3) are constructed in terms of electron fermion creation and annihilation operators. The concept of semiconjugacy is defined and shown to reduce the number of states that must be explicitly calculated. The states of the d shell are calculated and tabulated. In Chapter II it is shown how to extract n-body cfp's associated with arbitrary auxiliary quantum numbers from the n-body generalisation of Redmond's formula. The method is applied to give explicit formulae for the squares of one body cfp's of the atomic d-shell.
Group theory is applied in Chapter III to extend the quasiparticle formalism developed by Armstrong and Judd to expose the complete group structure of the eigenfunctions of the equivalent electron l shell. A simple method for relating quasiparticle states to determinantal states and for calculating quasiparticle matrix elements is developed. The need for fractional parentage coefficients in calculating these matrix elements is eliminated. In Chapter IV the technique and formalism is extended to describe general mixed configurations.
The hydrogen atom is factorised according to the scheme 0(4,2) 0(2,1) x 0(3) in Chapter V and the radial group 0(2,1) studied. It is shown that rkD n/(n+q) , where Da is a dilatation operator, is proportional to a tensor operator in this scheme, allowing a group theoretical study of the radial matrix element rk, including an explanation of the Pasternack and Sternheimer selection rule. The technique is extended in Chapter VI to solve a differential equation directly related to the generalised Kepler equation of Infeld and Hull in an 0(2,1) x 0(3) group scheme. This equation contains as special cases the Schrodinger, Klein-Gordan, and Dirac (two forms) hydrogen atoms. A generalised Pasternack and Sternheimer selection rule exists and some matrix elements can be evaluated group theoretically.