We have seen earlier
that we can represent the angular momentum states using the quantum numbers *j* and *m*_{j}
Now, suppose we have two sources of angular momentum represented by
and
*J*. How may we represent the composite state?

It follows from the commutation relations for angular momentum that two representations are possible:

Uncoupled representation:
< *j*_{1}*m*_{j1}*j*_{2}*m*_{j2} >

Coupled representation:
< *j*_{1}*j*_{2}*jm*_{j} >

*j*_{1}, *j*_{2}, *m*_{j1}, *m*_{j1} are the quantum numbers for the two angular momenta observables,
and
.

*j* and *m*_{j} are the corresponding quantum numbers of the composite angular momentum where

The quantum number *j* is obtained from the so-called Clebsch-Gordan series:
and the quantum number *m*_{j} is obtained as follows:

*m*_{j} = - *j* to + *j* in integral steps.
Similarly, *m*_{j1} = - *j*_{1} to + *j*_{1} in integral steps and *m*_{j2} = - *j*_{2} to + *j*_{2} in integral steps.

The two representations can be visualized pictorially using the vector-model of coupled angular momentum.

First we show the uncoupled representation below in the form of an animation.

Look carefully at the animation. The two blue arrows attached to the sides of the two cones represent the two angular momentum vectors
and
. The red arrow represents the total angular momentum .
||,
||, the z components *J*_{1z} and *J*_{2z} respectively, all remain fixed. However || and its z component *J*_{z} do change. Hence in this representation, the corresponding quantum numbers
*j*_{1}, *j*_{2}, *m*_{j1}, *m*_{j1} may be specified together but not with *j*, *m*_{j}.

Next we show the coupled representation below in the form of an animation.

Look carefully again at the animation. The two blue arrows attached to the sides of the two cones represent the two angular momentum vectors
and
as before. So does the red arrow represent the total angular momentum . However note now that the two cones are now tilted to the vertical z-axis and have as their common axis.
||,
|| remain fixed but the corresponding z components *J*_{1z} and *J*_{2z} vary. However || and the z component *J*_{z} remain fixed.
and
remain locked together, the angle between them remaining constant. Hence in this representation, the corresponding quantum numbers
*j*_{1}, *j*_{2}, *j*, *m*_{j} may be specified together but not with *m*_{j1} and *m*_{j2} . Also note that although *m*_{j1} and *m*_{j2} vary, their sum, *m*_{j} remains fixed.

Quantization of angular momentum has many interesting fall outs as we shall see subsequently.

2007-09-27