For 2 Hermitian operators A and B using the Cauchy-Schwarz inequality and assuming the expectation values of A and B are zero I get

(ΔA)

^{2}(ΔB)

^{2}≥ (1/4)|<(AB+BA)>|

^{2}+ (1/4)|<(AB-BA)>|

^{2}

Now both terms on the RHS are positive so why is this inequality usually just written with only the commutator term , dropping the anti-commutator term ? Surely it could be written with only the 1st term instead ? And why is it not normally written with both terms ?

Thanks