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Adding Two Spin-1/2 Particles
When adding two spin-1/2 particles, the total spin can be found using the rules of quantum mechanics. Here’s a brief overview of the process:
Individual Spins
Each particle has a spin quantum number .
Product basis for operator :
operator is diagonalisable in the product basis.
R. Shankar's book denotes these as :
Possible Combinations for total spin
When combining two spin-1/2 particles, the possible total spin quantum numbers are given by the rules of angular momentum addition:
- (triplet state, total spin 1)
- (singlet state, total spin 0)
Triplet State
When the total spin , there are three possible projections of the total spin along -axis labelled by :
- (both particles spin up, )
- (one particle spin up, one particle spin down, )
- (both particles spin down, )
Singlet State
When the total spin , there is only one possible projection:
- (one particle spin up, one particle spin down, )
So, when combining two spin-1/2 particles, you get one singlet state (total spin 0) and three triplet states (total spin 1).
Coupled basis
Total spin operator becomes diagonal in the coupled basis.
The coupled basis states are linear combinations of product basis states. In the present case, they are :