13.4 Particles in Magnetic Fields

Maxwell’s equations are fun, but back to real quantum mechanics.
The serious question in this section is how a magnetic field

Well, if the Hamiltonian (13.2) for a charged particle
is written out and cleaned up, {D.73}, it is seen that a
constant magnetic field adds two terms. The most important of the two
is

In terms of classical physics, this can be understood as follows: a
particle with angular momentum

In electromagnetics, the effective magnetic strength of a circling
charged particle is described by the so called orbital “magnetic dipole moment”

(13.37) |

The scalar part of the magnetic dipole moment, to wit,

used. Here T stands for Tesla, the kg/C-s unit of magnetic field strength.

Please, all of this is serious; this is not a story made up by this
book to put physicists in a bad light. Note that the original formula
had four variables in it:

The big question now is: since electrons have spin, build-in angular
momentum, do they still act like little magnets even if not going
around in a circle? The answer is yes; there is an additional term in
the Hamiltonian due to spin. Astonishingly, the energy involved pops
out of Dirac's relativistic description of the electron,
{D.74}. The energy that an electron picks up in a
magnetic field due to its inherent spin is:

It should be noted that really the

You might think that the above formula for the energy of an electron
in a magnetic field should also apply to protons and neutrons, since
they too are spin

Do note that due to the much larger mass of the proton, its actual
magnetic dipole moment is much less than that of an electron despite
its larger

For the neutron, the charge is zero, but the magnetic moment is not,
which would make its

At the start of this subsection, it was noted that the Hamiltonian for
a charged particle has another term. So, how about it? It is called
the “diamagnetic contribution,” and it is given by

The diamagnetic contribution can usually be ignored if there is net
orbital or spin angular momentum. To see why, consider the following
numerical values:

The first number gives the magnetic dipole energy, for a quantum of angular momentum, per Tesla, while the second number gives the diamagnetic energy, for a Bohr-radius spread around the magnetic axis, per square Tesla.

It follows that it takes about a million Tesla for the diamagnetic energy to become comparable to the dipole one. Now at the time of this writing, (2008), the world record magnet that can operate continuously is right here at the Florida State University. It produces a field of 45 Tesla, taking in 33 MW of electricity and 4 000 gallons of cooling water per minute. The world record magnet that can produce even stronger brief magnetic pulses is also here, and it produces 90 Tesla, going on 100. (Still stronger magnetic fields are possible if you allow the magnet to blow itself to smithereens during the fraction of a second that it operates, but that is so messy.) Obviously, these numbers are way below a million Tesla. Also note that since atom energies are in electron volts or more, none of these fields are going to blow an atom apart.