Next: Calculation of Clebsch-Gordon Coefficients Up: Addition of Angular Momentum Previous: Commutation Rules

# Clebsch-Gordon Coefficients

As we have seen, the operator group , , , and is incompatible with the group , , , and . This means that if the system is in a simultaneous eigenstate of the former group then, in general, it is not in a simultaneous eigenstate of the latter. In other words, if the quantum numbers , , , and are known with certainty then a measurement of the quantum numbers and will give a range of possible values. We can use the completeness relation (6.20) to write

 (6.22)

Thus, we can write the eigenkets of the first group of operators as a weighted sum of the eigenkets of the second set. The weights, , are called the Clebsch-Gordon coefficients. If the system is in a state where a measurement of , and is bound to give the results , and , respectively, then a measurement of and will give the results and , respectively, with probability .

The Clebsch-Gordon coefficients possess a number of very important properties. First, the coefficients are zero unless

 (6.23)

To prove this, we note that

 (6.24)

Forming the inner product with , we obtain

 (6.25)

which proves the assertion. Thus, the -components of different angular momenta add algebraically. So, an electron in an state, with orbital angular momentum , and spin angular momentum , projected along the -axis, constitutes a state whose total angular momentum projected along the -axis is . What is uncertain is the magnitude of the total angular momentum.

Second, the coefficients vanish unless

 (6.26)

We can assume, without loss of generality, that . We know, from Equation (6.23), that for given and the largest possible value of is (because is the largest possible value of , etc.). This implies that the largest possible value of is (because, by definition, the largest value of is equal to ). Now, there are allowable values of , and allowable values of . Thus, there are independent eigenkets, , needed to span the ket space corresponding to fixed and . Because the eigenkets span the same space, they must also form a set of independent kets. In other words, there can only be distinct allowable values of the quantum numbers and . For each allowed value of , there are allowed values of . We have already seen that the maximum allowed value of is . It is easily seen that if the minimum allowed value of is then the total number of allowed values of and is . In other words [59],

 (6.27)

This proves our assertion.

Third, the sum of the modulus squared of all of the Clebsch-Gordon coefficients is unity: that is,

 (6.28)

This assertion is proved as follows:

where use has been made of the completeness relation (6.20).

Finally, the Clebsch-Gordon coefficients obey two recursion relations. To obtain these relations, we start from

 (6.29)

where , et cetera, Making use of the well-known properties of the ladder operators, , , and , which are specified by analogy with Equations (4.55)-(4.56), we obtain

Taking the inner product with , and making use of the orthonormality property of the basis eigenkets, we get the desired recursion relations:

It is clear, from the absence of complex coupling coefficients in the previous relations, that we can always choose the Clebsch-Gordon coefficients to be real numbers. This is convenient, because it ensures that the inverse Clebsch-Gordon coefficients, , are identical to the Clebsch-Gordon coefficients. In other words,

 (6.30)

The inverse Clebsch-Gordon coefficients are the weights in the expansion of the in terms of the :

 (6.31)

It turns out that the recursion relations (6.32), together with the normalization condition (6.28), are sufficient to completely determine the Clebsch-Gordon coefficients to within an arbitrary sign (multiplied into all of the coefficients). This sign is fixed by convention. [To be more exact, each Clebsch-Gordon sub-table associated with a specific value of (see later) is undetermined to an arbitrary sign. It is conventional to give the Clebsch-Gordon coefficient with the largest value of a positive sign.] The easiest way of demonstrating this assertion is by considering a specific example.

Next: Calculation of Clebsch-Gordon Coefficients Up: Addition of Angular Momentum Previous: Commutation Rules
Richard Fitzpatrick 2016-01-22