How do I prove this equation for any function $f$ (can this even be proven for an arbitrary function?), where $\hat a$ and $\hat a^{\dagger}$ are the creation and annihilation operators for the 1D harmonic oscillator?
$$\hat af(\hat a^{\dagger} \hat a) = f(\hat a^{\dagger} \hat a + 1) \hat a$$
I have tried expanding $f$ as a power series and using the commutation relation of $\hat a$ and $\hat a^{\dagger}$.
Best Answer
I think the equation really only makes sense if it's acting on some state, say $|n\rangle$. Then it \begin{align*} a f(a^\dagger a) |n\rangle = a f(n) |n\rangle = f(n) \sqrt{n}|n-1\rangle \end{align*} and \begin{align*} f(a^\dagger a + 1) a |n\rangle = f(a^\dagger a + 1) \sqrt{n}|n-1\rangle = f(n-1+1) \sqrt{n}|n-1\rangle \end{align*} and the expressions are equal. Informally I'd argue that you can first measure how many particles exist in that state and then destroy a particle or you can first destroy one but then you have to add 1 to the number of particles that you measured afterwards.