Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 5 / b

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- Solution b

Solution

The occupation state is an eigenstate of the projected Quantum Hamiltonian. Since

a_{j}^{†} a_{j}^{†} a_{j} a_{j} = N_{j} (N_{j} - 1)

,

E_{N_{0}, N_{1}} = E N_{1} + \frac{U}{4 π} [N_{0} (N_{0} - 1) + N_{1} (N_{1} - 1) + 4 N_{0} N_{1}] .

Equivalently, the bracket is

N (N - 1) + 2 N_{0} N_{1}

, where

N = N_{0} + N_{1}

; there is no self-interaction for a single particle.

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