Spin-Spin Relaxation in a Simple System

A system of two particles of spin ½ coupled by their dipolar interaction and in an arbitrary magnetic field is considered. It is shown that exact expressions for the energy levels of the system can be obtained. The density matrix is calculated exactly for the cases where a magnetic field is suddenly applied, parallel or perpendicular, to the line joining the dipoles. It is used in the evaluation of the magnetic moment of the system. Instead of there being a gradual approach to an equilibrium situation the magnetic moment varies harmonically in time. A partially successful attempt is made to calculate the density matrix when the functional dependence of the magnetic field on the time is more complicated than a step function. The exact calculations for this simple system are compared with the approximate calculations of Waller for a system of $N$ spins. It is pointed out that there may be no gradual approach to equilibrium in the $N$-spin system, either.