We see that all energy levels, except for (the
so-called ground state), contain several states. Physicists say that the
energy levels are *degenerate*; in particular, the degree of degeneracy
of level is
. Mathematically, the -th
eigenspace of the energy operator has dimension .

Moralists may abhor degeneracy, but physicists love it, for it adds spice and complexity. (Of course, the same may hold for moral degeneracy!) Multi-dimensional eigenspaces furnish room for interesting transformation groups. Physical applications of group representation theory really took off after the discovery of quantum mechanics.

© 2001 Michael Weiss