130_notes.dvi

If we derive the fields frompotentials,

B~ = ∇×~ A~

E~ = −∇~φ−^1

c

∂A

∂t

then the first two Maxwell equations are automatically satisfied. Applying the second two equations
we getwave equationsin the potentials.

−∇^2 φ−

1

c

∂

∂t

(∇·~ A~) = 4πρ

−∇^2 A~+

1

c^2

∂^2 A~

∂t^2

+∇~

(

∇·~ A~+^1

c

∂φ

∂t

)

=

4 π

c

J~

Thesederivations(see section 20.5.1)are fairly simple using Einstein notation.

The two results we want to use as inputs for our study of Quantum Physics are

• the classical gauge symmetry and


• the classical Hamiltonian.

The Maxwell equations are invariant under agauge transformationof the potentials.

A~ → A~−∇~f(~r,t)

φ → φ+

1

c

∂f(~r,t)

∂t

Note that when we quantize the field, the potentials will play the role that wave functions do for the
electron, so this gauge symmetry will be important in quantum mechanics. We can use the gauge
symmetry to simplify our equations. For time independent charge and current distributions, the
coulomb gauge,∇·~ A~= 0, is often used. For time dependent conditions, theLorentz gauge,
∇·~ A~+^1
c

∂φ
∂t= 0, is often convenient. These greatly simplify the above wave equations in an obvious
way.

Finally, the classicalHamiltonian for electrons in an electromagnetic fieldbecomes

H=

p^2

2 m

→

1

2 m

(

~p+

e

c

A~

) 2

−eφ

The magnetic force is not a conservative one so we cannot just adda scalar potential. We know that
there is momentum contained in the field so the additional momentum term, as well as the usual
force due to an electric field, makes sense. The electron generates an E-field and if there is a B-field
present,E~×B~gives rise to momentum density in the field. The evidence that this is the correct
classical Hamiltonian is that we canderive(see section 20.5.2)the Lorentz Force from it.

20.2 The Quantum Hamiltonian Including a B-field

We will quantize the Hamiltonian

H=

1

2 m

(

~p+

e

c

A~

) 2

−eφ