Since I can't see any reason any non-physicist would be interested in this, this writeup assumes you have a background in basic quantum physics.

Consider a stationary electron in an uniform magnetic field. From classical electromagnetic theory the interaction energy of the system is -μ.B, where u is the magnetic moment of the electron (due to its spin) and B is the magnetic field. We want to find how the spin of the electron evolves with time. This analysis can be found in any book on elementary quantum mechanics.

We can assume without losing generality that the magnetic field points in the z-direction. Thus the interaction energy can be written as -μzB = -(eB/mc)Sz, where e is the charge of an electron, m is the electron's mass, and Sz is the z-axis spin operator. The quantity in parenthesis, twice the Larmor frequency, will be replaced by Ω. The Hamiltonian becomes ΩSz.

The state of the electron can be generally written as a|+> + b|->. From the Schrodinger equation we find that the state of the system as a function of time is ae-iΩt/2|+> + be+iΩt/2|->.

As a specific example, assume the electron is initially in an eigenstate of Sx. The Sx has both a and b equal to 1/sqrt(2). The expectation value <Sx> is found to be (hbar/2)cosΩt. The expectation value <Sy> is (hbar/2)sinΩt. Thus we see that the spin precesses in the x-y plane with frequency Ω.