QuantumPhysics.dvi

The degree to which this is possible is expressed by the Heisenberg uncertainty relations.

We begin by defining the following two states,

|α〉 = Aφ|φ〉

|β〉 = Bφ|φ〉 (4.19)

The Schwarz inequality on these states|〈α|β〉|^2 ≤||α||^2 ||β||^2 implies

∣∣

∣∣〈AφBφ〉φ

∣∣

∣∣

2

≤〈A^2 φ〉φ〈Bφ^2 〉φ= (∆φA)^2 (∆φB)^2 (4.20)

where the last equality was obtained by using the definition of the quantum fluctuations ∆φA

and ∆φBfrom (4.17). On the lhs, we use the decomposition of the product ofoperators into

commutator and anti-commutator,

AφBφ=

1

2

[Aφ,Bφ] +

1

2

{Aφ,Bφ} (4.21)

It is standard that for self-adjoint operatorsAandB, the quantitiesi[A,B] and{A,B}are

both self-adjoint. Hence we have

∣∣

∣∣〈AφBφ〉φ

∣∣

∣∣

2

=

1

4

∣∣

∣∣〈φ|[Aφ,Bφ]|φ〉

∣∣

∣∣

2

+

1

4

∣∣

∣∣〈φ|{Aφ,Bφ}|φ〉

∣∣

∣∣

2

(4.22)

Using the earlier Schwarz inequality for the above relation in which we drop the anti-

commutator, and taking the square root gives the inequality,

1

2

∣∣

∣∣〈[A,B]〉φ

∣∣

∣∣≤(∆φA) (∆φB) (4.23)

This inequality is the most general form of the Heisenberg uncertainty relations.

Examples: A familiar example is whenAandBare positionxand momentump

operators, satisfying [x,p] =i ̄hIH. The resulting uncertainty relation is well-known,

1

2

̄h≤∆x∆p (4.24)

A less familiar example is provided by angular momentum [Jx,Jy] =ihJ ̄ zand cyclic permu-

tations. Let|φ〉=|j,m〉with−j≤m≤j, then we have

1

2

|m|≤(∆φJx)(∆φJy) (4.25)

In this case, the uncertainty relation clearly depend on the state|φ〉, and quantum fluctua-

tions grow withm. In the special states wherem= 0, the operatorsJxandJymay actually

be observed simultaneously, since there 0≤(∆φJx)(∆φJy) even though the operatorsJxand

Jydo not commute on all states.