1.28 Addition of Angular Momentum
It is often required to add angular momentum from two (or more) sources (See section 21) 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 alsoeigenstates
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,L~ 1 and
L~ 2 to get a total angular momentumJ~. We will show that the total angular momentum quantum
number takes on every value in the range
|ℓ 1 −ℓ 2 |≤j≤ℓ 1 +ℓ 2.We can understand this qualitatively in thevector modelpictured below. We are adding two
quantum vectors.
1
l 1 +l 2
l 2
l 2
l
l 1 l^2
l 2
The length of the resulting vector is somewhere between the difference of their magnitudes and the
sum of their magnitudes, since we don’t know which direction the vectors are pointing.
The states of definite total angular momentum with quantum numbersjandm, can be written in
terms ofproducts of the individual states(like electron 1 is in this state AND electron 2 is in
that state). The general expansion is called theClebsch-Gordan series:
ψjm=∑
m 1 m 2〈ℓ 1 m 1 ℓ 2 m 2 |jmℓ 1 ℓ 2 〉Yℓ 1 m 1 Yℓ 2 m 2or in terms of the ket vectors
|jmℓ 1 ℓ 2 〉=∑
m 1 m 2〈ℓ 1 m 1 ℓ 2 m 2 |jmℓ 1 ℓ 2 〉|ℓ 1 m 1 ℓ 2 m 2 〉The Clebsch-Gordan coefficients are tabulated although we will compute many of them ourselves.
When combining states of identical particles, thehighest total angular momentum state,
s=s 1 +s 2 , will always besymmetric under interchange.The symmetry under interchange will
alternate asjis reduced.