21 Addition of Angular Momentum
Since total angular momentum is conserved in nature, we will find that eigenstates of the total
angular momentum operator are usually energy eigenstates. The exceptions will be when we apply
external Fields which break the rotational symmetry. We must therefore learn how to add different
components of angular momentum together. One of our first usesof this will be to add the orbital
angular momentum in Hydrogen to the spin angular momentum of the electron.
J~=~L+S~audio
Our results can be applied to the addition of all types of angular momentum.
This material is covered inGasiorowicz Chapter 15,inCohen-Tannoudji et al. Chapter X
and very briefly in Griffiths Chapter 6.
21.1 Adding the Spins of Two Electrons
The coordinates of two particles commute with each other: [p(1)i,x(2)j] = 0. They are inde-
pendent variables except that the overall wave functions for identical particles must satisfy the
(anti)symmetrization requirements. This will also be the case for the spin coordinates.
[S(1)i,S(2)j] = 0We define thetotal spin operators
S~=S~(1)+S~(2).Its easy toshow(see section 21.8.1)the total spin operators obey the same commutation relations
as individual spin operators
[Si,Sj] =i ̄hǫijkSk.
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This is a very important result since we derived everything about angular momentum from the
commutators. The sum of angular momentum will be quantized in the same way as orbital angular
momentum.
As with the combination of independent spatial coordinates, we canmakeproduct statesto
describe the spins of two particles. These products just mean, for example, the spin of particle 1 is
upandthe spin of particle 2 is down. There are four possible (product) spinstates when we combine
two spin^12 particles. Theseproduct states are eigenstates of totalSzbut not necessarily of
totalS^2. The states and theirSzeigenvalues are given below.
Product State TotalSzeigenvalue
χ(1)+χ(2)+ ̄h
χ(1)+χ(2)− 0
χ(1)−χ(2)+ 0
χ(1)−χ(2)− − ̄h