Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 33C / a

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- Solution a

Solution

Work in units with

ℏ = 1

. For unit mass and frequency, the two-dimensional isotropic harmonic oscillator has

H = \frac{1}{2} (p_{x}^{2} + x^{2}) + \frac{1}{2} (p_{y}^{2} + y^{2}) = a_{x}^{†} a_{x} + a_{y}^{†} a_{y} + 1.

The product number states are therefore energy eigenvectors:

∣ n_{x}, n_{y} ⟩, E_{n_{x}, n_{y}} = n_{x} + n_{y} + 1.

For the level

E = N + 1

, the possibilities are

(n_{x}, n_{y}) = (N, 0), (N - 1, 1), \dots, (0, N),

so its degeneracy is

N + 1 = E .

Restoring units multiplies every energy by

ℏ ω

.

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