Consider the free particle in one dimension: Ĥ = p̂²/2m. This Hamiltonian has both translational symmetry and inversion symmetry.
(a) Show that translations and inversion don’t commute.
(b) Because of the translational symmetry we know that the eigenstates of Ĥ can be chosen to be simultaneous eigenstates of momentum, namely fp (x) (Equation 3.32). Show that the parity operator turns fp(x) into f-p (x); these two states must therefore have the same energy.
(c) Alternatively, because of the inversion symmetry we know that the eigenstates of Ĥ can be chosen to be simultaneous eigenstates of parity, namely

Show that the translation operator mixes these two states together; they therefore must be degenerate.

Transcribed Image Text:

1 √πh COS px (P²) a 1 dsin (P). and