Consider a stationary spin-1/2 particle,
placed in a magnetic field
with magnitude pointing in the -direction.
In that case, its Hamiltonian is given by:
Where is the gyromagnetic ratio,
and is the Pauli spin matrix for the -direction.
Since is proportional to ,
they share eigenstates and .
The respective eigenenergies and are as follows:
Because is time-independent,
the general time-dependent solution is of the following form,
where and are constants,
and the exponentials are “twiddle factors”:
For our purposes, we can safely assume that and are real,
and then say that there exists an angle
satisfying and , such that:
Now, we find the expectation values of the spin operators
, , and .
The first is:
The other two are calculated in the same way,
with the following results:
The result is that the spin axis is off by from the -direction,
and is rotating (or precessing) around the -axis at the Larmor frequency :
- D.J. Griffiths, D.F. Schroeter,
Introduction to quantum mechanics, 3rd edition,