Might be simple but I couldn't see. We can easily derive momentum operator for position space by differentiating the plane wave solution. Analogously I want to derive the position operator for momentum space, however I am getting additional minus sign.(adsbygoogle = window.adsbygoogle || []).push({});

By replacing $$k=\frac{p}{\hbar}$$ and $$w=\frac{E}{\hbar}$$ into the plane wave solution, we get

$$\Psi=e^{ipx/\hbar-iEt/\hbar}$$

Then taking the derivative with respect to momentum,

$$\frac{\partial\Psi}{\partial p}=\frac{ix}{\hbar}\Psi$$

Then I get,

$$\hat{x}=-i\hbar \frac{\partial}{\partial p}$$

It has additional minus sign. Where is my mistake and/or how do I derive the position operator for momentum space in the simplest way?

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# The simplest derivation of position operator for momentum space

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