The group theory of the harmonic oscillator with applications in physics. (1972)
AuthorsHaskell, T. G.show all
The possibility of the group SU₃ being used in the description of the (d+s)N and (d+s)npm many-electron complexes is examined by symmetrization of the Coulomb Hamiltonian. By dividing the Coulomb interaction into symmetry conserving and symmetry violating terms it is found that while the SU₃ scheme tends to give a better description in the (d+s)N case it shows no improvement over the configurational scheme in the (d+s)npm complex. The scheme is, however, very useful for the calculation of matrix elements of operators normally found in atomic spectroscopy and a complete set of symmetrized , scalar, Hermitian spin-independent two particle operators acting within (d+s)npm configurations is constructed. The radial wavefunctions of the harmonic oscillator are found to form a basis for the representations of the group 0(2,1) in the group scheme Sp(6,R) ⊃ S0(3) x 0(2,1). The operators Tkp = r2k are shown to transform simply under the action of the group generators. The matrix elements of Tkq and a selection rule similar to that of Pasternack and Sternheimer are derived.
Finally the rich group structure of the harmonic oscillator is investigated and a dynamical group proposed which contains, as subgroups, the groups Sp(6,R), SU(3), H₄ and the direct product 0(2,1) x S0(3). Some remarks are made about contractions of groups, semidirect and direct products, and the generalization of the method to n-dimensions.