Past exam of the advanced quantum theory: path integrals and many-particle physics course of the University of Oxford / 2024 / 2 / a

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Solution

Let

V = L^{3}

be the box volume. The number operator is the sum of the occupation numbers of all momentum eigenstates, including the zero mode:

\hat{N} = \int_{V} d^{3} r ψ^{†} (r) ψ (r) = k \sum ψ_{k}^{†} ψ_{k} .

In the Fourier series for the contact interaction, the two creation operators supply phase

e^{- i (k_{1} + k_{2}) \cdot r}

and the two annihilation operators supply

e^{i (k_{3} + k_{4}) \cdot r}

. Fourier orthogonality with periodic boundary conditions gives

\int_{V} d^{3} r e^{i (k_{3} + k_{4} - k_{1} - k_{2}) \cdot r} = V δ_{k_{1} + k_{2}, k_{3} + k_{4}} .

The four normalization factors together contribute

V^{- 2}

, leaving the momentum-space contact-interaction Hamiltonian

H_{int} = \frac{u}{2 V} k_{1}, k_{2}, k_{3}, k_{4} \sum δ_{k_{1} + k_{2}, k_{3} + k_{4}} ψ_{k_{1}}^{†} ψ_{k_{2}}^{†} ψ_{k_{3}} ψ_{k_{4}} .

The Kronecker delta enforces momentum conservation. An equivalent expression that resolves the momentum transfer is

H_{int} = \frac{u}{2 V} \sum_{k, p, q} ψ_{k + q}^{†} ψ_{p - q}^{†} ψ_{p} ψ_{k} .

Both are already in normal ordering; no extra commutator constant is introduced.

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