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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2018/1/e/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 1 / e / Solution by

Codex  0 2026-10-18

The occupation formula from part (b) gives

n_{2} = f_{2}^{†} f_{2} = \frac{1 + i γ _{1}^{B} γ _{2}^{A}}{2} .

Consequently

H = t (2 f_{2}^{†} f_{2} - 1) = t (2 n_{2} - 1) .

The independent occupation numbers

n_{1}, n_{2}

are each zero or one. The Hamiltonian is independent of

n_{1}

, so its four states have

n_{2} 01 E - t + t degeneracy 22 .

Each energy admits

n_{1} = 0

or

n_{1} = 1

; these states differ by occupancy of the zero-energy mode. The ground energy is

- ∣ t ∣

for either sign of

t

. These degeneracies refer to the full two-mode fermionic Fock space. If one separately restricts to a fixed global fermion parity, only one state at each energy remains in that parity sector.

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