Degeneracy

We see that all energy levels, except for $n=1$ (the so-called ground state), contain several states. Physicists say that the energy levels are degenerate; in particular, the degree of degeneracy of level $n$ is $1+3+5+\ldots + (2n-1) = n^2$. Mathematically, the $n$-th eigenspace of the energy operator has dimension $n^2$.

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

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