130_notes.dvi

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,~pis the variable conjugate to~rand is related to the
velocity by

~p=m~v−

e

c

A~

as seen in our derivation of the Lorentz force (See Section 20.5.2).

Thecomputation(see section 20.5.3)yields

− ̄h^2

2 m

∇^2 ψ+

e

2 mc

B~·~Lψ+ e

2

8 mc^2

(

r^2 B^2 −(~r·B~)^2

)

ψ= (E+eφ)ψ.

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 rightside. We see two additional
terms due to the magnetic field. Anestimate(see section 20.5.4)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 2. 4 ×B 109 gauss compared to hydrogen binding energy), and the second canbe 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 (seeexample below).

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

HB=

e

2 mc

B~·L~=−~μ·B,~

where~μ=− 2 mce L~. This is, effectively, themagnetic momentdue to the electron’s orbital angular
momentum. In atoms, this term gives rise to theZeeman effect: otherwise degenerate atomic
states split in energy when a magnetic field is applied. Note that the electron spin which is not
included here also contributes to the splitting and will be studied later.

TheZeeman 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.

• See Example 20.4.1:Splitting of orbital angular momentum states in a B field.*

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

∆E=μBBmℓ

wheremℓis 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 becausethe typical radii can be much
bigger than in an atom. Aplasmais 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.