Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / b / iii

Table of contents
- Solution iii

Solution

Under unitary time evolution, each energy eigenstate acquires its own dynamical phase. Since

E_{2} = E_{- 2} = E_{⋆}

, the wavefunction is

Ψ (t, x) = \frac{2}{3} ϕ_{0} (x) - \frac{e ^{- i E_{⋆} t /ℏ}}{6} (ϕ_{2} (x) + ϕ_{- 2} (x)) = \frac{1 - e ^{- i Ω t} cos ( 2 x / R )}{3 π R}, Ω = \frac{E _{⋆}}{ℏ} = \frac{2ℏ}{m R ^{2}} .

At

t = 0

, the identity

1 - cos (2 x / R) = 2 sin^{2} (x / R)

recovers the normalized initial wavefunction.

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