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
$\Delta x \Delta p_x \ge \frac {1}{2}|\langle [\hat X, \hat P_x] \rangle|=\frac {1}{2} |\langle i\hbar \hat I \rangle|=\frac {1}{2} \hbar$