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ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2017/1/b/ii/solution

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Past exam of the quantum theory course of the University of Oxford / 2017 / 1 / b / ii / Solution by

Codex  0 2026-10-31

The requested energy eigenvalue is

E_{2}

. By the Born rule, its probability amplitude is the inner product

c_{2} = ⟨ ψ_{2}, Ψ (0)⟩

. Put

u = π x / a

and use

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

:

c_{2} = \frac{2}{π} \int_{0}^{π} sin 2 u e^{i u} sin u d u = \frac{1}{π} \int_{0}^{π} sin^{2} 2 u d u + \frac{2 i}{π} \int_{0}^{π} sin 2 u sin^{2} u d u = \frac{1}{2} .

The first integral is

π /2

; the second is zero under

u \mapsto π - u

. Therefore

Pr (E = E_{2}) = ∣ c_{2} ∣^{2} = \frac{1}{4} .

For the time-independent Hamiltonian operator, expand in its orthonormal basis of energy eigenstates:

Ψ (x, t) = \sum_{n \geq 1} c_{n} e^{- i E_{n} t /ℏ} ψ_{n} (x) .

Unitary time evolution changes

c_{2}

to

c_{2} e^{- i E_{2} t /ℏ}

, whose modulus is unchanged. Thus the same energy measurement probability holds at every time, even though relative quantum phases and other measurements can change.

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