 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2023/1/b/v/solution

Let

Π_{⋆} = ∣ ϕ_{2} ⟩ ⟨ ϕ_{2} ∣ + ∣ ϕ_{- 2} ⟩ ⟨ ϕ_{- 2} ∣

E_{⋆}

Ψ (T^{+}, x) = \frac{Π _{⋆} Ψ ( T ^{-} , x )}{1/3} = - \frac{e ^{- i E_{⋆} T /ℏ}}{2} (ϕ_{2} (x) + ϕ_{- 2} (x)) = - \frac{e ^{- i E_{⋆} T /ℏ}}{π R} cos (\frac{2 x}{R}) .

The factor

- e^{- i E_{⋆} T /ℏ}

cos (2 x / R) / π R

is an equivalent representative. The projective measurement preserves the quantum coherence between the two momentum eigenstates; it does not select one of them. This matters for the subsequent position probabilities.

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