For position \(\hat{x}\) and momentum \(\hat{p}_x\) operators the canonical commutation relation \({\left[\hat{x},\hat{p}_x\right]=i\hbar}\) implies that for any state \(\phi\) the standard deviation of position \(\Delta_{\phi}\hat{x}\) and the standard deviation of momentum \(\Delta_{\phi}\hat{p}_x\) are related by the inequality
$$\Delta_{\phi}\hat{x}\Delta_{\phi}\hat{p}_x\geq\frac{\hbar}{2}$$
Proof:
$$\left[\hat{A},\hat{B}\right]=i\hat{C}$$
An average value of \(\hat{D}^\dagger\hat{D}\), where \(\hat{D}=\hat{A}-\langle\hat{A}\rangle_{\phi}+i\alpha \left(\hat{B}-\langle\hat{B}\rangle_{\phi}\right)\), \(\alpha\in\mathrm{R}\):
$$\langle\hat{D}^\dagger\hat{D}\rangle_{\phi}=\lVert\hat{D}\phi \rVert^2=\left(\Delta_{\phi}\hat{A}\right)^2+\alpha^2\left(\Delta_{\phi}\hat{B}\right)^2-\alpha \langle\hat{C}\rangle_{\phi} \geq 0$$
$$\Delta_{\phi}\hat{A}\Delta_{\phi}\hat{B}\geq\frac{1}{2}|\langle\hat{C}\rangle_{\phi}|$$
If \(\hat{A}=\hat{x}\), \(\hat{B}=\hat{p}_x\) and \(\hat{C}=\hbar\) $$\Delta_{\phi}\hat{x}\Delta_{\phi}\hat{p}_x\geq\frac{\hbar}{2} $$