Aspects of Degeneracy

Abstract

Without symmetry, degeneracies are considered to be ‘accidential’, reflecting the fact that for a typical Hamiltonian Ĥ, representing a bound quantal system, no two of the energy levels E_n will coincide. But just as with road accidents the chance of a degeneracy can rise from negligible to inevitable if instead of considering individual Hamiltonians one embeds Ĥ in a population smoothly parameterised by variables ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}} = (X,Y,Z....)$. In section 2 I review an old argument of Von Neumann and Wigner (1929) indicating that for typical families $\hat{H}({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}})$ of real Hamiltonians, two parameters are necessary to produce a degeneracy, while if $\hat{H}({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}})$ is Hermitian (and not real) three parameters are necessary.