130_notes.dvi

We exploit the gauge symmetry in EM to show that, infield free regions, the functionfcan be
simply equal to a line integral of the vector potential (if we pick the right gauge).

f(~r) =

∫~r

~r 0

d~r·A.~

We use this to show that the magnetic flux enclosed by a superconductor is quantized.

We also show that magnetic fields can be used to change interference effects in quantum mechan-
ics. TheAharanov B ̈ohm Effectbrings us back to the two slit diffraction experiment but adds
magnetic fields.

electron 






flux

screen

gun

The electron beams travel through two slits in field free regions butwe have the ability to vary a
magnetic field enclosed by the path of the electrons. At the screen, the amplitudes from the two
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

∫

1

d~r·A~

ψ 2 → ψ 2 e

−i ̄hce

∫

2

d~r·A~

ψ =

(

ψ 1 e−i

ehc ̄Φ

+ψ 2

)

e

−ihc ̄e

∫

2

d~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.