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Let $H= \frac{p^2}{2m}$, then I am supposed to calculate $[x,e^{-iHt}]$.

My idea was to use $[x,p^n]=i \hbar n p^{n-1}$ and so I ended up by using the series for the exponential function with $-\frac{t \hbar}{m} e^{-iHt}$.

Could anybody tell me, whether this result is correct?

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Yes. Also note that in the momentum representation, $x = i\hbar \frac{d}{dp}$, which is what your commutation relation proved as a special case. You could use this shortcut right off the bat.

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