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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/j2/b/solution

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Past exam of the options II course of the University of Oxford / 2013 / J2 / b / Solution by

Codex  0 2026-10-31

Put

γ = mω /ℏ

. Substitution of

ψ = F (y) e^{- γ y^{2} /2}

in the one-dimensional Schrödinger equation with energy

(n + 1/2) ℏ ω

gives

F^{''} - 2 γ y F^{'} + 2 γn F = 0.

For degrees zero, one and two, coefficient comparison yields, up to independent nonzero constants,

F_{0} = 1, F_{1} = y, F_{2} = y^{2} - \frac{1}{2 γ} .

For the shifted coordinates

y_{i} = x_{i} - (r_{0})_{i}

, let

G (y) = e^{- γ ∣ y ∣^{2} /2}

. A basis of wave functions for the first three energy levels is:

$N = 0$ : $G$ .
$N = 1$ : $y_{1} G, y_{2} G, y_{3} G$ .
$N = 2$ : $(y_{i}^{2} - 1/ (2 γ)) G (i = 1, 2, 3), y_{1} y_{2} G, y_{1} y_{3} G, y_{2} y_{3} G$ .

These are products of the one-dimensional polynomial solutions. Arbitrary linear combinations within each listed eigenspace are also stationary states. Their counts

1, 3, 6

agree with part (a), so the degeneracies have all been included.

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