=

= .
Now, here's the trick.
By "x" and "p", I do *not* mean the position and momentum operators!
If X and P are the position and momentum operators,
then x := X -

.
Note that [x,p] = i, just like [X,P];
note that x and p are Hermitian, just like X and P;
in fact, everything I did with x and p was perfectly valid.
So that last inequality, >= 1/4, is just what we want.
BTW, if psi is Gaussian, = 1/4,
so this is the strongest inequality possible.