 Solution

ID: past-exam-of-the-quantum-matter-course-of-the-university-of-oxford/2023/1/b/solution

\hat{ψ} (r) = n \sum ϕ_{n} (r) \overset{a}{^}_{n}, \hat{ψ}^{†} (r) = n \sum ϕ_{n}^{*} (r) \overset{a}{^}_{n}^{†} .

The complex conjugation in the creation operator expansion is essential for complex orbitals. Indeed,

\overset{a}{^}_{n}^{†} = \int d^{3} r ϕ_{n} (r) \hat{ψ}^{†} (r)

creates the wavefunction

ϕ_{n}

[\hat{ψ} (r), \hat{ψ}^{†} (r^{'})] = \sum_{n} ϕ_{n} (r) ϕ_{n}^{*} (r^{'}) = δ^{(3)} (r - r^{'}),

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