# Why is the anticommutator of the uncertainty principle omitted if it serves to increase the accuracy of our "knowledge" of a quantum state?

The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous.

$$\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle [ A,B ] \rangle \vert^{2} + \dfrac{1}{4} \vert \langle \{ \Delta A, \Delta B \} \rangle \vert^{2}$$

On the right hand side, both quantities are real and positive. Sakurai argues that if you omit the anticommutator (as is typically done) the inequality is still true since the right hand side gets even smaller. However, this is bothersome since doesn't it mean that the square of the dispersion can approach even smaller values that allowed by the more rigorous uncertainty?

If I naively think about a number line, remove the anticommutator just lowers the floor for how much we can know about a quantum object doesn't it? Apologies for the bad drawing

• You do understand the entropic UP is a tighter bound, no? Oct 3, 2020 at 17:07

The anticommutator does not have immediate algebraic properties and no obvious physical interpretation beyond the “brute force” one of $$\hat A\hat B+\hat B\hat A$$.
For instance, whereas $$[L_x,L_y]=i\hbar L_z$$ the anticommutator is just the symmetrized product $$L_xL_y+L_yL_x$$ which isn’t anything special in the theory of angular momentum.