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I'm struggling trying to prove $[\overrightarrow{x}, f(\overrightarrow{p})]= i\hbar\frac{\partial f(\overrightarrow{p})}{\partial \overrightarrow{p}}$.

I already proved that $[x_i,p_i^n]=i\hbar \frac{\partial p_i^n}{\partial p_i}$ and I need to express $f$ in a power series but I don't know how to do that and derive the result.

Can someone help me?

Qmechanic
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Abel Gutiérrez
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1 Answers1

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I will not provide the full derivation, but will help you start your journey.

You can use the power series:

\begin{equation} f(p) = c^{(0)} + c^{(1)}p \;+\; ...c^{(n)}\frac{p^n}{n!} \end{equation}

Start looking at the commutator:

\begin{equation} \left[x, c^{(0)} + c^{(0)}p \;+\; ...\frac{c^{(n)}}{n!}p^n\right] \end{equation}

maybe look at it like:

\begin{equation} \sum_n \left[x, \frac{c^{(n)}}{n!}p^n\right] \end{equation}

user18764
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