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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2022/paper-1/section-b/b7/d/solution

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Past exam of the physics A course of the University of Cambridge / 2022 / Paper 1 / Section B / B7 / d / Solution by

Codex  0 2026-10-18

Identical spin one-half particles are fermions. By fermionic exchange symmetry, the complete spatial-and-spin wavefunction must change sign when the two particle labels are exchanged. Define the normalized antisymmetric spin-one-half singlet state

χ_{S} = \frac{∣ ↑ ⟩ _{1} ∣ ↓ ⟩ _{2} - ∣ ↓ ⟩ _{1} ∣ ↑ ⟩ _{2}}{2},

and the three symmetric spin-one-half triplet states

χ_{+} = ∣ ↑ ⟩_{1} ∣ ↑ ⟩_{2}, χ_{0} = \frac{∣ ↑ ⟩ _{1} ∣ ↓ ⟩ _{2} + ∣ ↓ ⟩ _{1} ∣ ↑ ⟩ _{2}}{2}, χ_{-} = ∣ ↓ ⟩_{1} ∣ ↓ ⟩_{2} .

The Hamiltonian is independent of spin, so these spin functions carry no additional energy.

Both particles can occupy

u_{0}

if their spin factor is antisymmetric. The unique ground state wavefunction is

Ψ_{g} (ϕ_{1}, ϕ_{2}) = u_{0} (ϕ_{1}) u_{0} (ϕ_{2}) χ_{S} = \frac{1}{2 π} χ_{S}, E_{g} = 0, g_{g} = 1.

Two equal-spin particles could not share this orbital, by the Pauli exclusion principle.

At the first excited energy,

E = ϵ = ℏ^{2} / (2 m R^{2})

, one orbital is

u_{0}

and the other is

u_{s}

with

s = + 1

or

- 1

. Define symmetric and antisymmetric normalized spatial functions

Φ_{s}^{\pm} (ϕ_{1}, ϕ_{2}) = \frac{u _{0} ( ϕ _{1} ) u _{s} ( ϕ _{2} ) \pm u _{s} ( ϕ _{1} ) u _{0} ( ϕ _{2} )}{2} = \frac{e ^{i s ϕ_{2}} \pm e ^{i s ϕ_{1}}}{2 π 2} .

For each

s

, all allowed wavefunctions are spanned by

Φ_{s}^{+} χ_{S}, Φ_{s}^{-} χ_{+}, Φ_{s}^{-} χ_{0}, Φ_{s}^{-} χ_{-}, s = \pm 1.

There are

2 (1 + 3)

orthonormal states, hence first-excited-state degeneracy $g = 8$ . In each product, exactly one factor is antisymmetric, ensuring the required total fermionic exchange symmetry.

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