Question: Consider the free particle in one dimension: H = p 2 /2m. This Hamiltonian has both translational symmetry and inversion symmetry. (a) Show that translations

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

1 h COS px (P) a 1 dsin (P). and

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

1 h COS px (P) a 1 dsin (P). and

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