Overview
Equation
{% \sigma_A = \langle (A - \langle A \rangle) \psi \; | \; (A - \langle A \rangle) \psi \rangle %}
{% = \langle \psi | (AB - A \langle B \rangle - B \langle A \rangle + \langle A \rangle \langle B \rangle) \psi \rangle %}
{% = \langle A B \rangle - \langle B \rangle \langle A \rangle - \langle A \rangle \langle B \rangle + \langle A \rangle \langle B \rangle %}
{% = \langle AB \rangle - \langle A \rangle \langle B \rangle %}
then
{% = \langle A B \rangle - \langle B \rangle \langle A \rangle - \langle A \rangle \langle B \rangle + \langle A \rangle \langle B \rangle %}
{% = \langle AB \rangle - \langle A \rangle \langle B \rangle %}
{% \langle (A - \langle A \rangle) \psi \; | \; (A - \langle A \rangle) \psi \rangle - \langle (B - \langle B \rangle) \psi \; | \; (B - \langle B \rangle) \psi \rangle %}
{% = \langle [A,B] \rangle %}
From the Schwartz inequality, we get
{% = \langle [A,B] \rangle %}
{% \sigma_A^2 \sigma_B^2 \geq \langle (A - \langle A \rangle) \psi \; | \; (B - \langle B \rangle) \psi \rangle %}
{% \sigma_A ^2 \sigma_B ^2 \geq (\frac{1}{2i} \langle [A,B] \rangle)^2 %}