Semiclassical Symmetries

Stephen C. Creagh

The author discusses semiclassical approximations that are adapted to given symmetry classes in quantum mechanics. Arbitrary abelian symmetries and also rotational symmetry are treated. Semiclassical approximations are derived for the projected propagator and energy dependent Green's function associated with a given irreducible representation of the symmetry group. From these they derive trace formulae, analogous to the usual trace formula, that determine the energy levels in a given symmetry class in terms of classical orbits.

Heinz Dietrich Doebner, V. K. Dobrev

Sudeshna Sinha, V B Sheorey

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.

Valter Moretti

Christian Grosche, Frank Steiner

J G Leopold, Ian Colin Percival, D Richards

The primitive semiclassical (EBK) theory of bound states is applied to finite non-relativistic systems with a discrete symmetry, particularly systems of identical particles. The symmetry properties of invariant tori are obtained, and illustrated for simple models with a pair of identical particles, each with one freedom. Examples without and with interaction between the particles, and with vibration and rotation are all considered. Particular care has to be taken with interacting particles that rotate around a ring.