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(A2), we can define the standard deviation of A in the state y, usually written DA even though it depends on y, by

(DA)2 = y, A2 y- y, Ay2 .
Note that this corresponds to the usual definition of standard deviation because y, Ay 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 non-commuting observables A and B:

Uncertainty Principle. Let A and B be self-adjoint operators on the Hilbert space H, and suppose y D(A2)D(B2)D(AB)D(BA) is a unit vector. Then

DADB 1
2
|y, [A,B]y|.

Proof - First we note that it suffices to show this for

A' = A - y, Ay
and
B' = B - y, By
instead. This is because [A', B'] = [A,B] and
y, A' y = y, Ay-y, y, Ay y = 0,
hence
(DA)2
= y, A'2 y
= y, (A2 - 2y, AyA + y, Ay2)y
= y, A2y- y,Ay2
= (DA)2,
and similarly for B.

We have

|y, [A',B']y|
= |A'y,B'y- B'y, A'y|
= 2 | Im A'y, B'y|
2| A'y,B'y|
2||A'y||||B'y||,
the last step using Cauchy-Schwartz; but
||A'y|| = y, A'2y1/2 = DA'
and similarly for B', so
|y, [A',B']y| 2 DADB,
as was to be shown.


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