Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Appendix C
Composition of angular
momenta: the coupled
representation
Let us consider a system in which J 1 and J 2 are two mutually commuting angular
momentum operators. This may happen when they refer to independent
physical systems: e.g. angular momenta of two different particles, or orbital
and spin angular momentum of the same particle. Let us also assume that
there are no interactions coupling them. We have four commuting observables
that describe the system: J1 2, J 1z, J2
2 and J 2z. The common eigenstates are
characterized by the set of quantum numbers: j 1 , m 1 , j 2 and m 2 . For fixed j 1
and j 2 , we have thus (2j 1 + 1)(2j 2 + 1) distinct states.
There is also another useful set of observables that describes the same system.
We define the operator total angular momentum
J = J 1 + J 2
(C.1)
It is immediate to verify that also J satisfies to the commutation algebra of the
angular momentum:
[J x , J y ] = [J 1x + J 2x , J 1y + J 2y ]
= [J 1x , J 1y ] + [J 2x , J 2y ]
= i¯hJ 1z + i¯hJ 2z
= i¯hJ z
(C.2)
since we have assumed that the commutators between components relative to
different systems are zero. Thus [J z , J 2 ] = 0.
We can then describe the system using the four operators J 2 1 , J 2 2 , J 2 and J z .
This is known as the coupled representation. In order to demonstrate that all
these four operators commute, we just need to show that [J 2 1 , J 2 ] = [J 2 2 , J 2 ] = 0
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