Transformation matrix between uncoupled and coupled representation for a two-spin system.

U2C = cgmatrix(S1,S2)
U2C = cgmatrix(S1,S2,Stot)
[U2C,Smtot] = ...
[U2C,Smtot,m12] = ...

cgmatrix calculates the transformation matrix needed to transform state vectors and operators from the uncoupled spin representation to the coupled spin representation (and vice versa), for a spin system with two spins.

S1 and S2 are the spin quantum numbers of the two spins.

If the third input Stot is given, only coupled basis states with the given total spin are included.

U2C is the transformation matrix from the uncoupled to the coupled basis. Its elements are Clebsch-Gordan coefficients. Each row is a coupled basis state represented in the uncoupled basis, and each column is an uncoupled basis state represented in the coupled basis.

The output Smtot contains a list of the quantum numbers S_tot and m_Stot of the coupled basis. Similarly, m12 contains a list of the quantum numbers m₁ and m₂ of the uncoupled basis.

The basis states in the uncoupled spin basis are Zeeman product states of the two spins, |S₁,m_S1,S₂,m_S2>, or |m_S1,m_S2> in abbreviated form. They are ordered in the standard descending lexicographical way as in the rest of EasySpin: top level is m_S1 in decreasing order, and then m_S2 in decreasing order. For example, for S1=1 and S2=1/2, the order of uncoupled states |m_S1,m_S2> is |1,1/2>, |1,-1/2>, |0,1/2>, |0,-1/2>, |-1,1/2>, |-1,-1/2>.

The coupled basis is in terms of the Zeeman states of the total spin with the quantum numbers S_tot and m_Stot, |S₁,S₂,S_tot,m_Stot> or |S_tot,m_Stot> for short. The total spin S_tot can assume values between S₁+S₂ and |S₁-S₂|. The basis states are ordered in descending order of S_tot, and for each value of S_tot, in descending order of m_Stot. For example, for S1=1 and S2=1/2 the possible values for the total spin S_tot are 3/2 and 1/2. The ordering of |S_tot, m_Stot> is |3/2,3/2>, |3/2,1/2>, |3/2,-1/2>, |3/2,-3/2>, |1/2,1/2>, |1/2,-1/2>.

To transform a state vector psi from the uncoupled to the coupled representation or vice versa, use

psi_c = U2C*psi_u;   % uncoupled -> coupled

C2U = U2C';
psi_u = C2U*psi_c;   % coupled -> uncoupled

To transform an operator matrix A (a spin operator, the spin Hamiltonian, or the density matrix) from the uncoupled to the coupled representation or vice versa, use

A_c = U2C*A_u*U2C';   % uncoupled -> coupled

C2U = U2C';
A_u = C2U*A_c*C2U';   % coupled -> uncoupled

Two spin-1/2 couple together to give a triplet (S_tot=1, three states with m_Stot = +1, 0, -1) and a singlet (S_tot=0, one state with m_Stot = 0). To calculate the representation of the triplet state |S_tot,m_Stot> = |1,0> in the uncoupled basis, we first define it as a vector in the coupled basis:

psi_c = [0;1;0;0];             % represents |1,0> in the coupled basis

Next, we obtain the transformation matrix using cgmatrix and apply it to obtain the representation of the state in the uncoupled representations:

U2C = cgmatrix(1/2,1/2)        % transformation matrix from uncoupled to coupled basis

U2C =
    1.0000         0         0         0
         0    0.7071    0.7071         0
         0         0         0    1.0000
         0    0.7071   -0.7071         0

psi_u = U2C'*psi_c

psi_u =
         0
    0.7071
    0.7071
         0

This shows that the coupled triplet state |1,0> is a linear combination of the uncoupled states |+1/2,-1/2> and |-1/2,+1/2>.

To determine the singlet state resulting from the coupling of two triplet spins represented in the uncoupled basis, use

[singlet_u,Sm,m12] = cgmatrix(1,1,0)

singlet_u =
          0         0    0.5774         0   -0.5774         0    0.5774         0         0
Sm =
     0     0
m12 =
     1     1
     1     0
     1    -1
     0     1
     0     0
     0    -1
    -1     1
    -1     0
    -1    -1

This also returns the uncoupled basis quantum numbers. It can be seen that the singlet state is a linear combination of the three states with m₁+m₂=0.

cgmatrix calls clebschgordan to calculate the matrix elements of the transformation matrix.

clebschgordan, wigner3j, wigner6j, wignerd