### D.19 The generalized uncertainty relationship

This note derives the generalized uncertainty relationship.

For brevity, define and , then the general expression for standard deviation says

Hermitian operators can be taken to the other side of inner products, so

Now the Cauchy-Schwartz inequality says that for any and ,

(See the notations for more on this theorem.) Using the Cauchy-Schwartz inequality in reversed order, you get

Now by the definition of the inner product, the complex conjugate of is , so the complex conjugate of is , and averaging a complex number with minus its complex conjugate reduces its size, since the real part averages away, so

The quantity in the top is the expectation value of the commutator . Writing it out shows that .