Heisenberg's Uncertainty from Dirac's Brackets
John Baez
Heisenberg's Uncertainty from Dirac's Brackets
by John Baez
Given an observable
A on
H and a state
y in
D(A^{2}),
we can define
the standard deviation of A in the state y, usually written
DA even though it depends on y, by
(DA)^{2} = áy, A^{2} yñ áy, Ayñ^{2} . 

Note that this corresponds to the usual definition of standard deviation
because áy, Ayñ is the mean value of the observable
A in the state y, as described in section 5.
The uncertainty principle gives a lower bound on the product
DADB for noncommuting observables A and B:
Uncertainty Principle. Let A and B be selfadjoint
operators on the Hilbert space H, and suppose
y Î D(A^{2})ÇD(B^{2})ÇD(AB)ÇD(BA) is a unit vector. Then
DADB ³ 
1
2

áy, [A,B]yñ. 

Proof  First we note that it suffices to show this for
and
instead. This is because [A', B'] = [A,B] and
áy, A' yñ = áy, Ayñáy, áy, Ayñ yñ = 0, 

hence


 
= áy, (A^{2}  2áy, AyñA + áy, Ayñ^{2})yñ 
 
= áy, A^{2}yñ áy,Ayñ^{2} 
 

 

and similarly for B.
We have


= áA'y,B'yñ áB'y, A'yñ 
 
 
 

 

the last step using CauchySchwartz; but
A'y = áy, A'^{2}yñ^{1/2} = DA' 

and similarly for B', so
áy, [A',B']yñ £ 2 DADB, 

as was to be shown.
File translated from T_{E}X by T_{T}H, version 0.9.