Using $[AB,C]=A[B,C]+[A,C]B$ and the canonical commutator,
$$
[\mathbf P,a_{\mathbf q}^\dagger]
=\int\frac{d^3p}{(2\pi)^3}\mathbf p\,
a_{\mathbf p}^\dagger[a_{\mathbf p},a_{\mathbf q}^\dagger]
=\boxed{\mathbf q\,a_{\mathbf q}^\dagger}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 1 / ii / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-301/1/ii/b/solution