operator $\hat A$ is a mathematical rule that when applied to a ket $\hat A|\phi\rangle$ transforms it into another ket $\hat A|\phi '\rangle $ and too for bra.
$\langle \phi| \hat A|\phi\rangle$ for short $\langle\hat A\rangle$
- what is the square of item $\langle\hat A\rangle^2$ ?
- what is the expectation value item $\langle\hat A^2\rangle$?
- what is difference between them$\Delta A=\sqrt{\langle\hat A^2\rangle-\langle\hat A\rangle^2}$?
Uncertainty Relation Between Two Operators
An interesting application of the commutator algebra is to derive a general relation giving the uncertainties product of two operator $\hat A$ and $\hat B$
if $\langle\hat A\rangle$ and $\langle\hat B\rangle$ be the expectation valus of to hermitian operators $\hat A$ and $\hat B$ with respect to a normalized state vector $|\psi\rangle$. that is, $\langle \psi| \hat A|\psi\rangle=\langle\hat A\rangle$ and $\langle \psi| \hat B|\psi\rangle=\langle\hat B\rangle$
The uncertainties $\delta A$ and $\delta B$ are defined by:
$$\delta A=\sqrt {\langle\hat A^2\rangle - \langle\hat A\rangle^2}$$, $$\delta B=\sqrt {\langle\hat B^2\rangle - \langle\hat B\rangle^2}$$, $$\delta A \delta B\ge \frac {1}{2}|\langle [\hat A, \hat B] \rangle|$$
leads to Heisenberg Uncertainty Relations
$$\Delta x \Delta p_x \ge \frac {1}{2}|\langle [\hat X, \hat P_x] \rangle|=\frac {1}{2} |\langle i\hbar \hat I \rangle|=\frac {1}{2} \hbar$$