130_notes.dvi

18.4 Anℓ= 1 System in a Magnetic Field*

We will derive the Hamiltonian terms added when an atom is put in a magnetic field in section 20.
For now, we can be satisfied with the classical explanation that the circulating current associated
with nonzero angular momentum generates amagnetic moment,as does a classical current loop.
This magnetic moment has the same interaction as in classical EM,

H=−~μ·B.~

For theorbital angular momentumin a normal atom, the magnetic moment is

~μ=

−e

2 mc

~L.

For the electron mass, in normal atoms, the magnitude of~μis oneBohr magneton,

μB=

e ̄h

2 mec

If we choose the direction ofBto be thezdirection, then themagnetic moment term in the
Hamiltonianbecomes

H=

μBB

̄h

Lz=μBB





1 0 0

0 0 0

0 0 − 1



.

So the eigenstates of this magnetic interaction are the eigenstates ofLzand theenergy eigenvalues
are +μBB, 0, and−μBB.

• See Example 18.10.6:The energy eigenstates of anℓ= 1 system in a B-field.*


• See Example 18.10.8:Time development of a state in a B field.*

18.5 Splitting the Eigenstates with Stern-Gerlach

A beam of atoms can be split into the eigenstates ofLzwith aStern-Gerlach apparatus. A
magnetic moment is associated with angular momentum.

~μ=

−e

2 mc

~L=μB

~L

̄h

This magnetic moment interacts with an external field, adding a termto the Hamiltonian.

H=−~μ·B~

If the magnetic field has a gradient in the z direction, there is a forceexerted (classically).

F=−

∂U

∂z

=μz

∂B

∂z

A magnet with astrong gradient to the fieldis shown below.