The Quantum Hamiltonian Including a B-field

We will quantize the Hamiltonian

in the usual way, by replacing the momentum by the momentum operator, for the case of a constant magnetic field.

Note that the momentum operator will now include momentum in the field, not just the particle's momentum.
As this Hamiltonian is written,
is the variable conjugate to
and is related to the velocity by

as seen in our derivation of the Lorentz force.

The
computation
yields

The usual kinetic energy term, the first term on the left side, has been recovered. The standard potential energy of an electron in an Electric field is visible on the right side. We see two additional terms due to the magnetic field. An estimate of the size of the two B field terms for atoms shows that, for realizable magnetic fields, the first term is fairly small (down by a factor of compared to hydrogen binding energy), and the second can be neglected. The second term may be important in very high magnetic fields like those produced near neutron stars or if distance scales are larger than in atoms like in a plasma (see example below).

So, for atoms, the dominant additional term is the one we anticipated classically
in section 18.4,

where . This is, effectively, the

The **Zeeman effect**, neglecting electron spin, is particularly simple to calculate because the the
hydrogen energy eigenstates are also eigenstates of the additional term in the Hamiltonian.
Hence, the correction can be calculated exactly and easily.

* Example:
Splitting of orbital angular momentum states in a B field.*

The result is that the shifts in the eigen-energies are

where is the usual quantum number for the z component of orbital angular momentum. The Zeeman splitting of Hydrogen states, with spin included, was a powerful tool in understanding Quantum Physics and we will discuss it in detail in chapter 23.

The additional magnetic field terms are important in a plasma because the typical radii can be much bigger
than in an atom.
A **plasma** is composed of ions and electrons, together to make a (usually) electrically neutral mix.
The charged particles are essentially free to move in the plasma.
If we apply an external magnetic field, we have a quantum mechanics problem to solve.
On earth, we use plasmas in magnetic fields for many things, including nuclear fusion reactors.
Most regions of space contain plasmas and magnetic fields.

In the example below, we will solve the Quantum Mechanics problem two ways:
one using our new Hamiltonian with B field terms,
and the other writing the Hamiltonian in terms of A.
The first one will exploit both
**rotational symmetry about the B field direction and
translational symmetry along the B field direction**.
We will turn the radial equation into the
**equation we solved for Hydrogen**.
In the second solution, we will use
**translational symmetry along the B field direction
as well as translational symmetry transverse** to the B field.
We will now turn the remaining 1D part of the Schrödinger equation into the 1D
**harmonic oscillator equation**,
showing that the two problems we have solved analytically are actually related to each other!

* Example:
A neutral plasma in a constant magnetic field.*

The result in either solution for the eigen-energies can be written as

which depends on 2 quantum numbers. is the conserved momentum along the field direction which can take on any value. is an integer dealing with the state in x and y. In the first solution we understand in terms of the radial wavefunction in cylindrical coordinates and the angular momentum about the field direction. In the second solution, the physical meaning is less clear.

Jim Branson 2013-04-22