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$\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$
\begin{eqnarray} (\Delta X)^2_\psi
&=& \langle (\hat{X} -\overline{X})^2 \rangle_\psi \label{EQ06}\\
&=& \average{\psi}{(\hat{X}- \overline{X})^2}\label{EQ07} \\
&=& \average{\psi}{ \hat{X}^2 -\hat{X}\overline{X}-\overline{X}\hat{X} + \overline{X}^2}\label{EQ08}\\
&=& \average{\psi}{\hat{X}^2 - 2\overline{X} \hat{X} +\overline{X}^2}\label{EQ09}\\
&=& \average{\psi}{\hat{X}^2} -\average{\psi}{2\overline{X} \hat{X}} + \average{\psi}{\overline{X}^2} \label{EQ10}\\ &=&\average{\psi}{\hat{X}^2} - 2\overline{X}\average{\psi}{\hat{X}} +\overline{X}^2\norm{\psi} =\overline{(X^2)} - 2\overline{X}^2 + \overline{X}^2\label{EQ11}\\&=& \overline{(X^2)} - \overline{X}^2\label{EQ12}\end{eqnarray}
