Note on Spin Precession

Consider first a magnetized classical object spinning about its center of mass, with angular momentum  and parallel magnetic moment.  Now add a magnetic field , say in the z-direction. This will exert a torque , easily solved to find the angular momentum vector  precessing about the magnetic field direction with angular velocity of precession .

 

(Proof: from , take  Of course, dLz/dt = 0, since  is perpendicular to , which is in the z-direction.)

 

The exact same result comes from a quantum analysis: for , the Hamiltonian for the interaction with the magnetic field is , so the time development is

 

                                                            

with

                                                           

but this is exactly the rotation operator (as shown earlier) through an angle  -gBt about !

So  just as for the classical case.

 

Paramagnetic Resonance

Note that the spin precession frequency is independent of the angle of the spin to the field.  Consider how all this looks in a frame of reference which is itself rotating about the z-axis. Let’s call the magnetic field , because we will soon be adding another one. In the rotating frame, the observed precession frequency is , so there is a different effective field in the rotating frame. Obviously, if the frame rotates exactly at the precession frequency, spins pointing in any direction will remain at rest in that frame—there’s no effective field at all.

 

Now suppose we add a small rotating magnetic field in the x,y plane, so

 

 

and in the rotating frame

                                                          

 

Now, if we tune the angular frequency of the small rotating field so that it exactly matches the precession frequency in the original static magnetic field, all the magnetic moment will see in the rotating frame is the small field in the x-direction!  It will therefore precess about the x-direction at the slow angular speed gB1. 

 

If the spins are lined up preferentially in the z-direction by the static field, and the small oscillating field is switched on for a time such that gB1t = p/2, the spins will be preferentially in the y direction in the rotating frame, so in the lab they will be rotating in the x,y plane, and a coil will pick up an ac signal from the induced emf.