7.3: Total Angular Momentum
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In general, a particle may have both spin and orbital angular momentum. Since L and S have the same dimensions, we can ask what is the total angular momentum J of the particle. We write this as
J=L+S≡L⊗I+I⊗S,
which emphasizes that orbital and spin angular momentum are described in distinct Hilbert spaces.
Since [Li,Sj]=0, we have
[Ji,Jj]=[Li+Si,Lj+Sj]=[Li,Lj]+[Si,Sj]=iℏϵijkLk+iℏϵijkSk=iℏϵijk(Lk+Sk)=iℏϵijkJk
In other words, J obeys the same algebra as L and S, and we can immediately carry over the structure of the eigenvalues and eigenvectors from L and S.
In addition, L and S must be added as vectors. However, only one of the components of the total angular momentum can be sharp (i.e., having a definite value). Recall that l and s are magnitudes of the orbital and spin angular momentum, respectively. We can determine the extremal values of J, denoted by ±j, by adding and subtracting the spin from the orbital angular momentum, as shown in Figure 3:
|l−s|≤j≤l+s.
For example, when l=1 and s=12, the possible values of j are j=12 and j=32.
The commuting operators for J are, first of all, J2 and Jz as we expect from the algebra, but also the operators L2 and S2. You may think that Sz and Lz also commute with these operators, but that it not the case:
[J2,Lz]=[(L+S)2,Lz]=[L2+2L⋅S+S2,Lz]=2[L,Lz]⋅S≠0
We can construct a full basis for total angular momentum in terms of J2 and Jz, as before:
J2|j,mj⟩=ℏ2j(j+1)|j,mj⟩ and Jz|j,mj⟩=mjℏ|j,mj⟩.
Alternatively, we can construct spin and orbital angular momentum eigenstates directly as a tensor product of the eigenstates
L2|l,m⟩|s,ms⟩=ℏ2l(l+1)|l,m⟩|s,ms⟩ and Lz|l,m⟩|s,ms⟩=mℏ|l,m⟩|s,ms⟩,
and
S2|l,m⟩|s,ms⟩=ℏ2s(s+1)|l,m⟩|s,ms⟩ and Sz|l,m⟩|s,ms⟩=msℏ|l,m⟩|s,ms⟩.
Since the Lz and Sz do not commute with J2, the states |j,mj⟩ are not the same as the states |l,m⟩|s,ms⟩.