130_notes.dvi

Now we willapply the differential operator to the exponentialto identify the new terms.
Note that∇~eiλ(~r,t)=eiλ(~r,t)i∇~λ(~r,t).

eiλ(~r,t)

(

̄h

i

∇~+e

c

A~+e

c

∆~A+ ̄h

(

∇~λ(~r,t)

))^2

ψ = eiλ(~r,t)

(

i ̄h

∂

∂t

+eφ+e∆φ− ̄h

∂λ(~r,t)

∂t

)

ψ

(

̄h

i

∇~+e

c

A~+e

c

∆~A+ ̄h

(

∇~λ(~r,t)

))^2

ψ =

(

i ̄h

∂

∂t

+eφ+e∆φ− ̄h

∂λ(~r,t)

∂t

)

ψ

Its easy to see that we canleave this equation invariant with the following choices.

∆~A = − ̄hc

e

~∇λ(~r,t)

∆φ =

̄h

e

∂λ(~r,t)

∂t

We can argue thatwe need Electromagnetism to give us the local phase transformation
symmetryfor electrons. We now rewrite the gauge transformation in the more conventional way,
the convention being set before quantum mechanics.

ψ(~r,t) → eiλ(~r,t)ψ(~r,t)

A~ → A~−∇~f(~r,t)

φ → φ+

1

c

∂f(~r,t)

∂t

f(~r,t) =

̄hc

e

λ(~r,t).

20.5.8 Magnetic Flux Quantization from Gauge Symmetry

We’ve shown that we can compute the functionf(~r) from the vector potential.

f(~r) =

∫~r

~r 0

d~r·A~

A superconductor excludes the magnetic field so we have our field free region. If we take a ring of
superconductor, as shown, we get a condition on the magnetic fluxthrough the center.