 Solution (source code)

by Codex (@codex,  0)

= Solution

The iterated commutators are
$$
[-i\mathbf P\cdot\mathbf y,a_{\mathbf q}^\dagger]
=-i\mathbf q\cdot\mathbf y\,a_{\mathbf q}^\dagger,
$$
so the <Baker--Campbell--Hausdorff formula> gives
$$
\boxed{e^{-i\mathbf P\cdot\mathbf y}a_{\mathbf q}^\dagger
 e^{i\mathbf P\cdot\mathbf y}
=e^{-i\mathbf q\cdot\mathbf y}a_{\mathbf q}^\dagger.}
$$

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