Answers to Problem Set 2

Prob 1

Let us introduce the notatiom (Subscript[Y, lm])(θ,ϕ) ⇒ Y[l,m,θ,ϕ] and (Subscript[χ, μ]) ⇒ χ[μ]. With this notation, we may write the wave function for a state (j,m) obtained by combining orbital angular moment 1 and spin angular momentum 1/2 as

Ψ[j_, m_]  = 
      Sum[ClebschGordan[{1, m - μ}, {1/2, μ}, {j, 
              m}]  χ[μ]  Y[1, 
            m - μ, θ, ϕ], {μ, -j, j}]

There are two possibilities, j = 1/2 and j = 3/2. For the first case (j=1/2) the two substates are:

Table[Ψ[1/2, m], {m, -1/2, 1/2}]

For the second case (j=3/2) the 4 substates are:

Table[Ψ[3/2, m], {m, -3/2, 3/2}]

Prob 2

Combining nuclear spin state  Θ[m]  ( I= 9/2) and atomic angular momentum state   ψ[μ]  ( J = 1/2),
we obtain two possible values of F (4 & 5) with wavefunctions Ψ[F,M]. The degeneracies of these
states are 9 & 11, respectively. The corresponding wave functions are:

Ψ[F_, M_]  = 
      Sum[ClebschGordan[{9/2, M - μ}, {1/2, μ}, {F, 
              M}]  ψ[μ]  Θ[
            M - μ], {μ, -F - 1/2, F + 1/2}]

Table[Ψ[4, m], {m, -4, 4}]

Table[Ψ[5, m], {m, -5, 5}]

Expand[Ψ[4, 4] Ψ[5, 4]]