Quantum Mechanics for Engineers 

© Leon van Dommelen 

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 CauchySchwartz inequality says that for any and
,
(See the notations for more on this theorem.) Using the
CauchySchwartz 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
.