Spin-Dependent 1/r Potential

Consider two identical particles of mass m and spin 1/2. They interact only through the potential

(1)

where g > 0 and σ_j are Pauli spin matrices which operate on the spin of particle j.

a) Construct the spin eigenfunctions for the two particle states. What is the expectation value of V for each of these states?

b) Give the eigenvalues of all of the bound states.

SOLUTION

a) The spin operator is s = σ/2. For spin 1/2 the expression s . s = s(s+ 1)Ĩ = 3Ĩ/4 becomes, for Pauli matrices, σ . σ = 3Ĩ, where Ĩ is the unit matrix. The total spin operator for the two-particle system is

(1)

(2)

(3)

For the spin singlet state S = 0 then V_{S = 0} = -3g/r, while for the spin triplet state (S = 1), then V_{S = 1} = g/r.

b) The potential is repulsive for the triplet state, and there are no bound states. There are bound states for the singlet state since the potential is attractive. For the hydrogen atom the potential is –e²/r and the eigenvalues are

(4)

Our two-particle bound state has 3g instead of e² and the reduced mass m/2 instead of the mass m, so we have the eigenvalues

(5)