QuantumPhysics.dvi

Examples of vector observables include the angular momentum generatorsJthemselves, the

position operatorsx, and the momentum generatorsp. We shall now establish selection

rules forallvector observables.

We begin by decomposingVin the raising and lowering basis that we also use forJitself.

V 0 =V 3 V±=

1

√

2

(V 1 ±iV 2 ) (9.60)

so that the commutator relations become,

[J 3 ,Vq] = q ̄hVq q=−, 0 ,+

[J±,V 0 ] = ∓

√

2 ̄hV±

[J±,V±] = 0

[J±,V∓] = ±

√

2 ̄hV 0 (9.61)

In fact, we can write this set of commutation relations in a way that will easily generalize to

tensor observables later,

[Ja,Vq] =

∑

q′

(

D(1)(Ja)

)

q,q′Vq

′ (9.62)

whereq′ranges over the values−, 0 ,+.

9.9 Selection rules for vector observables

We shall study relations between various matrix elements ofVq, and obtain the following

results, which are often referred to as theWigner-Eckardt theorem,

(1) (m′−m−q)〈j′,m′,α′|Vq|j,m,α〉= 0 (9.63)

(2) 〈j′,m′,α′|Vq|j,m,α〉= 0 whenever |j′−j|≥ 2

(3) matrix elements withj′−j= 0,±1 are all related to one another;

Here,αandα′ represent all the quantum numbers other thanj andm. In the Coulomb

problem, for example,αwould be the principal quantum number, which fixes the energy of

the state. For the special vector observableJq=Vq, the matrix elements〈j′,m′,α′|Vq|j,m,α〉

will vanish unless we also haveα′=α, but for general vector observables, this need not be

the case.

• To prove (1), we take the matrix elements of [J 3 ,Vq] = ̄hqVq, or

〈j′,m′,α′|J 3 Vq−VqJ 3 |j,m,α〉− ̄hq〈j′,m′,α′|Vq|j,m,α〉= 0 (9.64)