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 = áy, A2 yñ- áy, Ayñ2 .
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 non-commuting observables A and B:

Uncertainty Principle. Let A and B be self-adjoint operators on the Hilbert space H, and suppose y Î D(A2)ÇD(B2)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

A' = A - áy, Ayñ
and
B' = B - áy, Byñ
instead. This is because [A', B'] = [A,B] and
áy, A' yñ = áy, Ayñ-áy, áy, Ayñ yñ = 0,
hence
(DA)2
= áy, A'2 yñ
= áy, (A2 - 2áy, AyñA + áy, Ayñ2)yñ
= áy, A2yñ- áy,Ayñ2
= (DA)2,
and similarly for B.

We have

|áy, [A',B']yñ|
= |áA'y,B'yñ- áB'y, A'yñ|
= 2 | Im áA'y, B'yñ|
£ 2| áA'y,B'yñ|
£ 2||A'y||||B'y||,
the last step using Cauchy-Schwartz; but
||A'y|| = áy, A'2yñ1/2 = DA'
and similarly for B', so
|áy, [A',B']yñ| £ 2 DADB,
as was to be shown.

File translated from TEX by TTH, version 0.9.