slits interfereψ=ψ 1 +ψ 2. Let’s start withB= 0 andA= 0 everywhere. When we change theB
field, the wavefunctions must change.
ψ 1 → ψ 1 e−i ̄hce∫
1d~r·A~ψ 2 → ψ 2 e−i ̄hce∫
2d~r·A~ψ =(
ψ 1 e−iehc ̄Φ
+ψ 2)
e−ihc ̄e∫
2d~r·A~The relative phase from the two slits depends on the flux between the slits. By varying theBfield,
we willshift the diffraction patterneven thoughB= 0 along the whole path of the electrons.
While this may at first seem amazing, we have seen similar effects in classical E&M with an EMF
induced in a loop by a changingBfield which does not touch the actual loop.
20.4 Examples
20.4.1 The Naive Zeeman Splitting
The additional term we wish to consider in the Hamiltonian is 2 μceB~·~L. Choosing the z axis so that
the constant field points in the z direction, we have
HZeeman=
eBz
2 μcLz.In general, the addition of a new term to the Hamiltonian will require usto use an approximation
to solve the problem. In this case, however, the energy eigenstates we derived in the Hydrogen
problem arestill eigenstates of the full HamiltonianH=Hhydrogen+HZeeman. Remember,
our hydrogen states are eigenstates of H,L^2 andLz.
(Hhydrogen+HZeeman)ψnℓm= (En+mμBB)ψnℓmThis would be a really nice tool to study the number of degenerate states in each hydrogen level.
When the experiment was done, things did not work our according toplan at all. The magnetic
moment of the electron’ s spin greatly complicates the problem. We will solve this later.
20.4.2 A Plasma in a Magnetic Field
An important place where both magnetic terms come into play is in aplasma. There, many
electrons are not bound to atoms and external Electric fields are screened out. Let’s assume there
is a constant (enough) B field in the z direction. We then have cylindrical symmetry and will work
in the coordinates,ρ,φ, andz.
− ̄h^2
2 me∇^2 ψ+eB
2 mecLzψ+e^2 B^2
8 mec^2(x^2 +y^2 )ψ= (E+eφ)ψThe problem clearly hastranslational symmetry along the z direction and rotational sym-
metry around the z axis. Given the symmetry, we know thatLzandpzcommute with the