復習計算メモ

\begin{equation}
\left[ \hat{p}, \hat{x} \right] = -i\hbar,\ \left[ \partial/\partial x, x \right] = 1
\end{equation}
\begin{eqnarray}
 & &  \langle x|  \left[ \hat{p}, \hat{x} \right] |\psi \rangle \\
 &=& \int dx'\langle x|  \left[ \hat{p}, \hat{x} \right] | x' \rangle \langle x'|\psi \rangle \\
&=&  \int dx' \int dx'' \left\{ \langle x| \hat{p} |x''\rangle \langle x''| \hat{x}   | x' \rangle
-  \langle x| \hat{x} |x''\rangle \langle x''| \hat{p}  | x' \rangle  \right\} \langle x'|\psi \rangle \\
&=& \int dx' \int dx'' \left\{ p(x,x'') x' \delta(x'' - x') - x'' \delta(x-x'')  p(x'',x') \right\}\psi(x') \\
&=&  \int dx' \{ p(x,x') x' - x p(x,x') \}\psi(x')\\
&=& -i\hbar \psi(x)\\
&=&   \langle x|  (-i\hbar) | \psi \rangle
\end{eqnarray}
where $p(x,x') = \langle x| \hat{p} | x' \rangle $.
\begin{equation}
 \therefore p(x,x') = -i\hbar\delta(x-x') \frac{\partial}{\partial x'},\ \langle x | \hat{p} |\psi \rangle = -i\hbar \frac{\partial \psi(x)}{\partial x}
\end{equation}
\begin{eqnarray}
&&  p\langle x | p \rangle \\
&=& \langle x |\hat{p}| p \rangle \\
&=& -i\hbar \partial/\partial x  \langle x | p \rangle
\end{eqnarray}
\begin{equation}
\therefore  \langle x | p \rangle = \exp\{ i x p /\hbar\}
\end{equation}