Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 15B / b / iii / Solution

The specified initial state is normalized. The requested energy is

E_{2}

, and its probability amplitude is

c_{2} = \int_{0}^{a /2} \frac{2}{a} sin \frac{2 π x}{a} \frac{2}{a} sin \frac{2 π x}{a}, d x = \frac{1}{2} .

Thus the Born rule gives

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

Unitary evolution changes each energy coefficient only by a phase, so this probability remains

1/2

at both

T

and

2 T

. If an energy measurement returns

E_{2}

, the state collapses to the nondegenerate eigenstate

χ_{2}

; a subsequent energy measurement therefore returns

E_{2}

with probability

1 .

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