= Solution
{c}
The <quantized field> expansion in the complete <orthonormal basis> of <orbitals> is
$$
\boxed{\hat\psi(\mathbf r)=\sum_n\phi_n(\mathbf r)\hat a_n,\qquad
\hat\psi^\dagger(\mathbf r)=\sum_n\phi_n^*(\mathbf r)\hat a_n^\dagger.}
$$
The complex conjugation in the <creation operator> expansion is essential for complex <orbitals>. Indeed, $\hat a_n^\dagger=\int d^3r\,\phi_n(\mathbf r)\hat\psi^\dagger(\mathbf r)$ creates the <wavefunction> $\phi_n$. The <resolution of the identity in an orthonormal basis> gives
$$
[\hat\psi(\mathbf r),\hat\psi^\dagger(\mathbf r')]
=\sum_n\phi_n(\mathbf r)\phi_n^*(\mathbf r')
=\delta^{(3)}(\mathbf r-\mathbf r'),
$$
recovering the <canonical commutation relation> and the spatial <Dirac delta function>.
Back to article page
