130_notes.dvi

In the example below, we will solve the Quantum Mechanics problem twoways: one using our new
Hamiltonian with B field terms, and the other writing the Hamiltonian in terms of A. The first
one will exploit bothrotational symmetry about the B field direction and translational
symmetry along the B field direction. We will turn the radial equation into theequation we
solved for Hydrogen. In the second solution, we will usetranslational symmetry along the B
field direction as well as translational symmetry transverseto the B field. We will now turn
the remaining 1D part of the Schr ̈odinger equation into the 1Dharmonic oscillator equation,
showing that the two problems we have solved analytically are actuallyrelated to each other!

• See Example 20.4.2:A neutral plasma in a constant magnetic field.*

The result in either solution for the eigen-energies can be written as

En=

eB ̄h

mec

(

n+

1

2

)

+

̄h^2 k^2

2 me

which depends on 2 quantum numbers. ̄hkis the conserved momentum along the field direction
which can take on any value.nis an integer dealing with the state in x and y. In the first solution
we understandnin terms of the radial wavefunction in cylindrical coordinates and the angular
momentum about the field direction. In the second solution, the physical meaning is less clear.

20.3 Gauge Symmetry in Quantum Mechanics

Gauge symmetry in Electromagnetism was recognized before the advent of quantum mechanics. We
have seen that symmetries play a very important role in the quantumtheory. Indeed, in quantum
mechanics, gauge symmetry can be seen as the basis for electromagnetism and conservation of charge.

We know that the all observables are unchanged if we make a global change of the phase of the
wavefunction,ψ→eiλψ. We could call thisglobal phase symmetry.All relative phases (say for
amplitudes to go through different slits in a diffraction experiment) remain the same and no physical
observable changes. This is a symmetry in the theory which we already know about. Let’s postulate
that there is a bigger symmetry and see what the consequences are.

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. Thislocal phase symmetryis bigger than the global one.

Its clear that this transformation leaves theabsolute square of the wavefunction the same,
but what about the Schr ̈odinger equation? It must also be unchanged. Thederivatives in the
Schr ̈odinger equationwill act onλ(~r,t) changing the equation unless we do something else to
cancel the changes.
1
2 m

(

~p+

e

c

A~

) 2

ψ= (E+eφ)ψ

A littlecalculation(see section 20.5.7) shows that the equation remains unchanged if we also
transform the potentials

A~ → A~−∇~f(~r,t)

φ → φ+

1

c

∂f(~r,t)

∂t

f(~r,t) =

̄hc

e

λ(~r,t).