Referrence to Modern Quantum Mechanics, J.J.Sakurai,
I found the "Generalized Uncertainty Principle" is that
<( s.d. of operator A)^2> <( s.d. of operator B)^2> >= 1/4 |<[A,B]>|^2 + 1/4 |<{s.d of A, s.d of B}>|^2
I hope it is not difficult to read, as I don't know how to type it formally...