 Solution

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

Using

sin^{2} u = \frac{1}{2} - \frac{1}{4} e^{2 i u} - \frac{1}{4} e^{- 2 i u}

, the energy eigenstate decomposition is

Ψ (0, x) = N 2 π R (\frac{1}{2} ϕ_{0} (x) - \frac{1}{4} ϕ_{2} (x) - \frac{1}{4} ϕ_{- 2} (x)) .

Only the energy eigenvalues zero and

E_{⋆} = 2 ℏ^{2} / (m R^{2})

occur. The last two terms are distinct momentum eigenstates in the same energy eigenspace.

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