 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-2/section-a-ii/14/ii/c/solution

Each energy component acquires its own phase:

ψ (x, t) = \frac{1}{S} [0.2 e^{- i ϵ t /ℏ} ψ_{1} (x) + 0.3 e^{- 4 i ϵ t /ℏ} ψ_{2} (x) + 0.4 e^{- 9 i ϵ t /ℏ} ψ_{3} (x) + 0.843 e^{- 16 i ϵ t /ℏ} ψ_{4} (x)] .

The

1/ S

factor normalizes the printed rounded input; it can be omitted if the coefficients are regarded as an already normalized rounded representation.

The energy probabilities and $⟨ E ⟩$ remain constant under this time-independent hamiltonian because phase factors have unit modulus. position probabilities can nevertheless vary: interference between different energy components generally changes with their relative phases. A stationary energy distribution is not the same thing as a stationary spatial probability distribution.

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