Adding Angular Momenta and the Coupled Basis
For angular-momentum quantum numbers j1 and j2, total J runs from |j1-j2| to j1+j2 in steps of one. Product and coupled bases span the same state space. Two-spin expansions describe singlet and triplet states without themselves causing energy splitting.
What a learner can do afterwards
- Lists the total angular momentum values available from two given ones
- Counts states in both bases and checks the totals agree
- Writes a coupled state as a combination of product states for a two-spin case
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For two angular momenta, total J runs from |j1-j2| to j1+j2 in steps of one. The absolute value makes the lower endpoint nonnegative. The product basis labels m1 and m2 separately. The coupled basis labels total J and M=m1+m2. These bases span the same state space; changing basis adds no physical states.
One angular momentum j has 2j+1 projection states. Pair every first-system state with every second-system state: the product dimension is (2j1+1)(2j2+1). In the coupled basis, each allowed J contributes 2J+1 states. Add these sector dimensions to obtain the full coupled dimension.
Two full bases of the same finite state space must have equal counts. Different counts mean at least one calculation or claim of completeness is incorrect. For j1=3/2 and j2=1/2, the product count is 4×2=8. The allowed totals J=1,2 contribute 3+5=8. The agreement checks these two descriptions of the same space.
To check a proposed list of total angular momenta, compute both endpoints from |j1-j2| and j1+j2. Reject entries outside this interval. Starting at the lower endpoint, also check unit spacing and that every allowed value is included. An entry above the sum cannot belong to the coupled space.
For two spins 1/2, write up and down as u and d in particle order. The triplet is |1,1>=uu, |1,0>=(ud+du)/sqrt(2), |1,-1>=dd. The singlet is |0,0>=(ud-du)/sqrt(2), up to overall phase. The triplet is symmetric under exchange; the singlet is antisymmetric. To convert a superposition to product states, substitute these formulas, distribute its coefficients and collect matching terms. The product states ud and du are orthonormal.
Energy comes from the Hamiltonian, not from the basis labels. If H=E I on a state space, then Hpsi=Epsi for every vector psi in it. Any coupled-basis vector has the same energy E. Relabelling the vectors as singlet and triplet cannot create a splitting. A splitting requires a Hamiltonian that distinguishes states or sectors.
J runs from |j1-j2| to j1+j2 by ones. Both bases span the same space. Two spins 1/2 give a triplet and a singlet; changing basis alone does not split energies.
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