20 Electrons in an Electromagnetic Field
In this section, we will study the interactions of electrons in an electromagnetic field. We will
compute the additions to the Hamiltonian for magnetic fields. The gauge symmetry exhibited in
electromagnetism will be examined in quantum mechanics. We will show that a symmetry allowing
us to change the phase of the electron wave function requires theexistence of EM interactions (with
the gauge symmetry).
These topics are covered inGasiorowicz Chapter 13,and inCohen-Tannoudji et al. Com-
plementsEV I,DV IIandHIII.
20.1 Review of the Classical Equations of Electricity and Magnetism in CGS Units
You may be most familiar with Maxwell’s equations and the Lorentz force law in SI units as given
below.
∇·~ B~ = 0
~∇×E~+∂B
∂t= 0
∇·~ E~ = ρ
ǫ 0∇×~ B~−^1
c^2∂E
∂t= μ 0 J~F~=−e(E~+~v×B~).These equations have needless extra constants (not) of naturein them so we don’t like to work in
these units. Since the Lorentz force law depends on the product of the charge and the field, there is
the freedom to, for example, increase the field by a factor of 2 butdecrease the charge by a factor
of 2 at the same time. This will put a factor of 4 into Maxwell’s equationsbut not change physics.
Similar tradeoffs can be made with the magnetic field strength and theconstant on the Lorentz force
law.
The choices made in CGS units are more physical (but still not perfect). There are no extra constants
other thanπ. Our textbook and many other advanced texts use CGS units and so will we in this
chapter.Maxwell’s Equationsin CGS units are
∇·~ B~ = 0
∇×~ E~+^1
c∂B
∂t= 0
~∇·E~ = 4πρ∇×~ B~−^1
c∂E
∂t=
4 π
cJ.~
TheLorentz Forceis
F~=−e(E~+^1
c~v×B~).In fact, an even better definition (rationalized Heaviside-Lorentzunits) of the charges and fields can
be made as shown in the introduction to field theory in chapter 32. For now we will stick with the
more standard CGS version of Maxwell’s equations.