Recent work on Lie’s method of extended groups to obtain symmetry groups and invariants of differential equations of mathematical physics is surveyed. As an essentially new contribution one-parameter Lie groups admitted by three-dimensional harmonic oscillator, three-dimensional wave equation, Klein-Gordon equation, two-component Weyl’s equation for neutrino and four-component Dirac equation for Fermions are obtained.

We investigate here the origin of nuclear levelwidth as an effect of the coupling of particle mode and the surface vibration mode of the nucleus. This interaction is taken to be stochastic in nature, characterized by a single correlation timet₀, the random nature of the interaction originating from the partition of the total hamiltonian into those of the two modes. The Redfield equation of motion for the density matrix for the particle mode is solved. The solution of the Redfield equation shows that the occupation number in any particle state decays with a time constant depending on the correlation timet₀ and the quantum-mechanical matrix elements of the interaction hamiltonian. The inverse of this decay time will give the width of this level. Numerical calculations have been done for₈₂²⁰⁷Pb₁₂₅.

Lie’s method of differential equation is used to obtain the one-parameter Lie groups admitted by the time-dependent Schrödinger 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 andN neutrons in a harmonic oscillator field, both isotropic and anisotropic, the r-parameter Lie groups are seraidirect 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(JV+1)+2].

Fokker-Planck type equations have been classified according to the groups of contact transformations to which they belong. It has been found that there are only five classes as in the case of groups of point transformations. We have also obtained the algebraic structures of the corresponding Lie algebras. However, there are isomorphies in their group properties. The corresponding basis sets of functionally independent invariants formed by the generators of these groups have also been obtained.