It is often required to add angular momentum from two (or more) sources together to get states of definite total angular momentum. For example, in the absence of external fields, the energy eigenstates of Hydrogen (including all the fine structure effects) are also eigenstates of total angular momentum. This almost has to be true if there is spherical symmetry to the problem.
As an example, lets assume we are adding the orbital angular momentum from two electrons,
to get a total angular momentum
We will show that the total angular momentum quantum number takes on every value in the range
The states of definite total angular momentum with quantum numbers
can be written in terms of products of the individual states
(like electron 1 is in this state AND electron 2 is in that state).
The general expansion is called the Clebsch-Gordan series:
When combining states of identical particles, the highest total angular momentum state, , will always be symmetric under interchange.The symmetry under interchange will alternate as is reduced.
The total number of states is always preserved.
For example if I add two
states together, I get total angular
momentum states with
There are 25 product states since each
state has 5 different possible
Check that against the sum of the number of states we have just listed.
We will use addition of angular momentum to:
Jim Branson 2013-04-22