 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2014/paper-1/5/d/solution

⟨ N_{1} (N_{1} - 1)⟩ = N (N - 1) p^{2}

, its mode-zero counterpart

N (N - 1) (1 - p)^{2}

, and

⟨ N_{0} N_{1} ⟩ = N (N - 1) p (1 - p)

. Consequently the exact projected expectation is

\frac{E ( χ )}{N} = E p + \frac{U ( N - 1 )}{4 π} [1 + 2 p (1 - p)] .

Since

p (1 - p) = sin^{2} χ /4

and

n = N / (2 π)

\frac{E ( χ )}{N} ≃ E sin^{2} (χ /2) + \frac{n U}{2} [1 + \frac{1}{2} sin^{2} χ] .

The finite-

N

correction is simply the replacement

N \mapsto N - 1

in the interaction term. Number correlations must be retained to obtain that correction.

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