 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-2/section-b/b9/solution

With angular-coordinate measure

d ϕ

, the normalized one-particle orbitals are

u_{l} (ϕ) = e^{i lϕ} / 2 π

. Single-valuedness requires integer

l

. Their products obey

\int_{0}^{2 π} \int_{0}^{2 π} ∣ ψ ∣^{2} d ϕ_{1} d ϕ_{2} = 1, E_{l_{1} l_{2}} = ϵ_{R} (l_{1}^{2} + l_{2}^{2}), ϵ_{R} = \frac{ℏ ^{2}}{2 m R ^{2}} .

If arc-length measure is used instead, each one-particle wavefunction normalization includes

R^{- 1/2}

; the physical energy is unchanged. For the following parts impose antisymmetry on the full spatial-and-spin state of the two fermions.

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