Two three-dimensional vectors are represented as two copies of the representation of

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

Two three-dimensional vectors are represented as two copies of the

j=1

representation of

\mathfrak{su}(2)

, with basis states

|1,1\rangle

|1,0\rangle

|1,-1\rangle

in each factor. The tensor product decomposes as

1\otimes1=0\oplus1\oplus2

, and the total lowering operator is

J_-=J_-^{(1)}+J_-^{(2)}

, which commutes with the exchange operator

P

that swaps the two factors. ((a)) Starting from the top state

|1,1\rangle\otimes|1,1\rangle

of the

j=2

multiplet, apply

J_-

once and show that the resulting

M=1

state is symmetric under

P

. ((b)) The two-dimensional space of

M=1

states is spanned by

|1,1\rangle\otimes|1,0\rangle

and

|1,0\rangle\otimes|1,1\rangle

. Identify the combination orthogonal to the result of part (a), give its symmetry under

P

, and explain why this fixes the symmetry of the entire

j=1

multiplet. ((c)) The

M=0

states are

|1,1\rangle\otimes|1,-1\rangle

|1,0\rangle\otimes|1,0\rangle

and

|1,-1\rangle\otimes|1,1\rangle

. Find the combination that

J_+=J_+^{(1)}+J_+^{(2)}

sends to zero, which is the

j=0

state, and give its symmetry under

P

. Hence state which ladders make up the six symmetric states, and so which pieces a symmetric rank-two Cartesian tensor carries.