Dipolar Interactions

Two spin-1/2 particles are separated by a distance a = aẑ and interact only through the magnetic dipole energy

(i)

where μ_j is the magnetic moment of spin j. The system of two spins consists of eigenstates of the total spin (S²) and total S_z.

a) Write the Hamiltonian in terms of spin operators.

b) Write the Hamiltonian in terms of S² and S_z.

c) Give the eigenvalues for all states.

SOLUTION

a) We assume the magnetic moment is a vector parallel to the spin with a moment μ_j = μ₀s_j where μ₀ is a constant. Then we write the Hamiltonian as

(1)

The second term contains only s_jz components since the vector a is along the z-direction.

b) We write

(2)

(3)

(4)

(5)

(6)

For s = 1/2 we have s(s + 1) = 3/4 and s²_jz = 1/4, so we can write

(7)

(8)

(9)

(10)

c) The addition of two angular momenta with s = 1/2 gives values of S which are 0 or 1:

For S = 1 there are three possible eigenvalues of S_z = (-1, 0, 1), which gives an energy of E₀ = (-1, 2, -1).

For S = 0 there is one eigenvalue of S_z = 0, and this state has zero energy.