# Heisenberg uncertainty made stronger by ommiting anticommutator

The following reasoning appears in Sakurai/Napolitano - Modern Quantum Mechanics, 3rd ed, ch 1:

Using a Schwarz inequality, it is proved that $$| \langle \Delta A \Delta B \rangle |^2 \geq \frac{1}{4} |\left \langle \right[A, B] \rangle |^2 + \frac{1}{4} |\left \langle \right \{ \Delta A, \Delta B \} \rangle|^2$$ Where $$\Delta A = A - \langle A\rangle$$ is some sort of dispersion/difference operator, $$\langle A \rangle$$ is expectation value, $$[A,B] = AB-BA$$ is the commutator and $$\left\{ P, Q \right\} = PQ + QP$$ is the anti-commutator. Assume $$A$$, $$B$$, $$\Delta A$$, $$\Delta B$$ are Hermitian (last two might be proved, but let's just assume).

Both terms are non-negative, therefore it is also true that: $$| \langle \Delta A \Delta B \rangle |^2 \geq \frac{1}{4} |\left \langle \right[A, B] \rangle |^2$$ and the author says this second version is stronger than the first.

My question is: how so? Isn't a met condition stronger depending on how restricting it is? The first equation is clearly more restrictive. Also, why don't we take the anti-commutator as the stronger version? Also, if the second equation is stronger than the first, why isn't $$| \langle \Delta A \Delta B \rangle |^2 \geq 0$$, which is trivially true taken as a even stronger condition (it isn't even useful, then)?

Edit: just found this What is the meaning of the anti-commutator term in the uncertainty principle?, but the direction is this question is opposite, since this book says literally: "The proof of is now complete because the omission of the second (the anticommutator) term of can only make the inequality relation stronger"

$$| \langle \Delta A \Delta B \rangle |^2 \geq \frac{1}{4} |\left \langle \right[A, B] \rangle |^2 + \frac{1}{4} |\left \langle \right \{ \Delta A, \Delta B \} \rangle|^2 \geq \frac{1}{4} |\left \langle \right[A, B] \rangle |^2 \geq 0.$$
• … and of course the commutator is also often an element in the same (Lie) algebra as $A$ and $B$, so one can often leverage all the Lie algebra machinery to some use. Mar 9, 2022 at 0:00