- #1

ergospherical

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Misplaced Homework Thread

What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*}

<x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\

&= \int dp' \ p' <x|p'> <p'|x'> \\

&= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\

&= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')}

\end{align*}

<x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\

&= \int dp' \ p' <x|p'> <p'|x'> \\

&= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\

&= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')}

\end{align*}