130_notes.dvi

0

path 1

path 2

B field

r r

Consider two different paths from~r 0 to~r.

f 1 (~r)−f 2 (~r) =

∮

d~r·A~=

∫

dS~·∇×~ A~=

∫

dS~·B~= Φ

The difference between the two calculations of f is the flux.

Nowfis not a physical observable so thef 1 −f 2 does not have to be zero, but,ψdoes have to be
single valued.

ψ 1 = ψ 2

⇒ e−i

̄hcef^1

=e−i

̄hcef^2

⇒

e

̄hc

(f 1 −f 2 ) = 2nπ

⇒ Φ =f 1 −f 2 =

2 nπ ̄hc

e

The flux is quantized.

Magnetic flux is observed to be quantized in a region enclosed by a superconductor. however, the
fundamental charge seen is 2e.

20.6 Homework Problems

1. Show that the HamiltonianH= 21 μ[~p+ecA~(~r,t)]^2 −eφ(~r,t) yields the Lorentz force law for an
   electron. Note that the fields must be evaluated at the position of the electron. This means
   that the total time derivative ofA~must also account for the motion of the electron.


2. Calculate the wavelengths of the three Zeeman lines in the 3d→ 2 ptransition in Hydrogen
   atoms in a 10^4 gauss field.


3. Show that the probability flux for system described by the Hamiltonian

H=

1

2 μ

[~p+

e

c

A~]^2