Subgroups of the Euclidean group and symmetry breaking in nonrelativistic quantum mechanics

Abstract

A systematic study of explicit symmetry breaking in the nonrelativistic quantum mechanics of a scalar and a spinor particle is presented. The free Schrödinger (or Pauli) equation is invariant under the Euclidean group E(3); an external field will break this symmetry to a lower one. We first find all continuous subgroups of E(3) and then for each subgroup construct the most general (within certain restrictions) external field that breaks the symmetry from E(3) to the corresponding subgroup. For a scalar particle the interaction term is assumed to be of the form V (r)+A(r)P, where P is the momentum, i.e., it involves an arbitrary scalar and vector potential. For a spinor particle it is of the form V (r)+A(r)P +B(r) σ+Mik(r) σiPk (σi are the Pauli matrices). A one-to-one correspondence between subgroups of E(3) and classes of ’’symmetry breaking potentials’’ is established. The remaining subgroup symmetry is then used to solve or at least simplify the obtained Schrödinger equation. The existence of a one-dimensional invariance group (for a particle in a field) leads to the partial separation of variables and determines the functional dependence of the wavefunction on one variable. A two-dimensional group implies the complete separation of variables and the functional dependence on two variables. A higher dimensional invariance group implies the separation of variables in one or more systems of coordinates and in some cases specifies the wavefunction completely.