 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s3/3/6/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S3 / 3 / 6 / Solution by

Codex  0 2026-10-25

Choose the infinite well as

0 < x < a

with real eigenfunctions

ψ_{n} = 2/ a sin (nπ x / a)

. This fixes the relative phase convention for the specified initial plus state. Each of the first two states has probability

1/2

in the left half. Their cross integral is

\int_{0}^{a /2} ψ_{1} ψ_{2} d x = \frac{2}{π} \int_{0}^{π /2} sin u sin 2 u d u = \frac{4}{3 π} .

The evolving superposition therefore gives

P_{left} (t) = \frac{1}{2} + \frac{4}{3 π} cos (\frac{( E _{2} - E _{1} ) t}{ℏ}) .

Since

E_{n} = n^{2} π^{2} ℏ^{2} / (2 m a^{2})

, the angular frequency and ordinary frequency are

ω = \frac{3 π ^{2} ℏ}{2 m a ^{2}}, f = \frac{ω}{2 π} = \frac{3 h}{8 m a ^{2}} .

The probability varies between

1/2 \pm 4/ (3 π)

and remains in

[0, 1]

. Changing the sign convention of one eigenfunction while keeping the same physical initial state requires changing its superposition coefficient too; otherwise one has prepared the other initial state and reversed the initial spatial bias.

Figure 1.
Original two-state transition and left-half well probabilities over two recurrence periods
. Details.

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