QuantumPhysics.dvi

Thus, it remains to relate the different matrix elements ofV^0 to one another. This is done

using the double commutator relation,

[J+,[J−,V 0 ]] = 2 ̄h^2 V 0 (9.73)

Taking the matrix elements of this relation,

〈j′,m′,α′|[J+,[J−,V 0 ]]|j,m,α〉= 2 ̄h^2 〈j′,m′,α′|V 0 |j,m,α〉 (9.74)

In view of (1), we know that only the matrix elements withm′=mcan be non-vanishing,

and we now specialize to this case without loss of generality. Working out the commutators

and applying the operatorsJ±to the states, we obtain,

(

Nj−′,mNj+′,m− 1 +Nj,m+Nj,m−+1− 2

)

〈j′,m,α′|V 0 |j,m,α〉 (9.75)

=Nj−′,mNj,m− 〈j′,m− 1 ,α′|V 0 |j,m− 1 ,α〉+Nj+′,mNj,m+〈j′,m+ 1,α′|V 0 |j,m+ 1,α〉

This is a second order recursion relation. Without loss of generality we shall assume that

j′≥j. Takingm=j, we get

(

Nj−′,jNj+′,j− 1 − 2

)

〈j′,j,α′|V 0 |j,j,α〉=Nj−′,jNj,j−〈j′,j− 1 ,α′|V 0 |j,j− 1 ,α〉 (9.76)

Given that we now have two initial data for the second order recursion relation, it fol-

lows that all matrix elements〈j′,m,α′|V 0 |j,m,α〉are known as a function of the single one

〈j′,j,α′|V 0 |j,j,α〉, which proves (3).

9.10 Tensor Observables

Higher rank tensor observables sometimes occur as well, but it is rare that tensors of rank

higher than 2 are needed, and we restrict here to tensors of rank2. Quadrupole moments

produce a tensor of rank 2 for example, xixj which is manifestly symmetric. Generally,

tensor of rank 2 arise as sums of tensor products of vector observables, as is the case in the

quadrupole moments. Consider a single one of these tensor products,

Tij=UiVj i,j= 1, 2 , 3 (9.77)

whereUiandVj are vector observables. Here we shall generally take the observablesUand

V to be different quantities, but they could be the same.

The tensorTijadmits a unique decomposition into its trace partT 0 , its anti-symmetric

partT 1 and its symmetric traceless partT 2 , according to the following formula,

Tij=

1

3

δijT^0 +

∑^3

k=1

εijkTk^1 +Tij^2 (9.78)