130_notes.dvi

ψ=v(x)eikyyeikzz

1

2 me

(

− ̄h^2

d^2

dx^2

+

(

eB

c

) 2 (

x+

̄hcky

eB

) 2 )

v(x) =

(

E−

̄h^2 kz^2

2 me

)

v(x)

− ̄h^2

2 me

d^2

dx^2

+

(

1

2

me

(

eB

mec

) 2 (

x+

̄hcky

eB

) 2 )

v(x) =

(

E−

̄h^2 kz^2

2 me

)

v(x)

This is thesame as the 1D harmonic oscillator equationwithω=meBecandx 0 =− ̄hckeB.

E=

(

n+

1

2

)

̄hω=

̄heB

mec

(

n+

1

2

)

+

̄h^2 k^2 z

2 me

So we get thesame energies with a much simpler calculation. The resultingstates are
somewhat strangeand are not analogous to the classical solutions. (Note that an electron could
be circulating about any field line so there are many possible states, just in case you are worrying
about the choice ofkyandx 0 and counting states.)

20.5.7 A Hamiltonian Invariant Under Wavefunction Phase (or Gauge)Transformations

tions

We want to investigate what it takes for theHamiltonian to be invariant under a local phase
transformationof the wave function.

ψ(~r,t)→eiλ(~r,t)ψ(~r,t)

That is, we can change the phase by a different amount at each pointin spacetime and the physics
will remain unchanged. We know that the absolute square of the wavefunction is the same. The
Schr ̈odinger must also be unchanged.

(

~p+

e

c

A~

) 2

ψ= (E+eφ)ψ

So let’spostulate the following transformationthen see what we need to keep the equation
invariant.

ψ(~r,t) → eiλ(~r,t)ψ(~r,t)

A~ → A~+∆~A

φ → φ+ ∆φ

We now need to apply this transformation to the Schr ̈odinger equation.

(

̄h

i

∇~+e

c

A~+e

c

∆~A

) 2

eiλ(~r,t)ψ=

(

i ̄h

∂

∂t

+eφ+e∆φ

)

eiλ(~r,t)ψ