I'm trying to find some information on how to add the angular momentum of three or more particles, but all the sources I look at deal with only two. In this case I understand that if the angular momentum numbers of the two particles are j1 and j2, then the possible total angular momentum numbers are J=(j1+j2),(j1+j2−1)+...+|j1−j2|. However, I don't see how to combine this to three particles.
For example, if I have three protons in a 1d5/2 nuclear energy level (for example), then the protons all have angular momentum j=5/2. However, how do I then find the possible total angular momentum of the state? I appreciate that the particles cannot occupy the same state, and hence must have different mj values which range from 5/2,3/2...−5/2, and then the mj is the sum of these, but then how could this be used to find the possible total angular momentum of the state (not just the total mj).
Answer
For every half-integer j=n/2,n∈Z, there is an irreducible representation of SU(2) Dj=exp(−i→θ⋅→J(j))
Suppose then we wish to find the angular momentum of a baryon. We need D1/2⊗D1/2⊗D1/2. Convince yourself of the following: Given three matrices A,B,C we have A⊗(B⊕C)=A⊗B⊕A⊗C
The obvious generalization of this is Dm⊗Dj⊗Dℓ=Dm⊗(j+ℓ⨁k=|j−ℓ|Dk)=j+ℓ⨁k=|j−ℓ|Dm⊗Dk=j+ℓ⨁k=|j−ℓ|m+k⨁n=|m−k|Dn
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