One-parameter Lie groups admitted by time-dependent Schroedinger equation: atoms, molecules and nucleons in harmonic oscillator field

Lie's method of differential equation is used to obtain the one-parameter Lie groups admitted by the time-dependent Schrodinger equations for atoms, molecules and nucleons in harmonic oscillator field. This group for atoms and molecules is isomorphic to 10-parameter inhomogeneous orthogonal group in 4 dimensions, irrespective of the numbers of nuclei and electrons. For Z protons and N neutrons in a harmonic oscillator field, both isotropic and anisotropic, the r-parameter Lie groups are semidirect products of an invariant subgroup and a factor group. In the case of isotropic oscillator field r is 1/2[3Z(3Z-1)+3N(3N-1)+2], while for the anisotropic oscillator field r is 1/2[3Z(Z+1)+3N(N+1)+2]. (author)