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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 4C / Solution

Separation with Dirichlet walls gives

Ψ_{n_{x} n_{y} n_{z}} = sin \frac{n _{x} π x}{a} sin \frac{n _{y} π y}{b} sin \frac{n _{z} π z}{c},

E_{n_{x} n_{y} n_{z}} = \frac{ℏ ^{2} π ^{2}}{2 m} (\frac{n _{x}^{2}}{a ^{2}} + \frac{n _{y}^{2}}{b ^{2}} + \frac{n _{z}^{2}}{c ^{2}}), n_{x}, n_{y}, n_{z} \geq 1.

For

a < b < c

, the ground state

(1, 1, 1)

has a nondegenerate energy. The cheapest excitation changes the quantum number in the longest direction, so the first excited state

(1, 1, 2)

also has a nondegenerate energy.

If

a < b = c

, the ground-state energy remains nondegenerate, but

(1, 2, 1)

and

(1, 1, 2)

have equal first-excited energy, giving degeneracy two.

Solved by gpt-5.6-sol high.

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