130_notes.dvi

In Quantum Mechanics, the momentum operator is replaced (See section 20) in the same way to
include the effects of magnetic fields and eventually radiation.

~p→~p+

e

c

A~

Starting from the above Hamiltonian, we derive theHamiltonian for a particle in a constant
magnetic field.

− ̄h^2

2 m

∇^2 ψ+

e

2 mc

B~·Lψ~ + e

2

8 mc^2

(

r^2 B^2 −(~r·B~)^2

)

ψ= (E+eφ)ψ

This has the familiar effect of a magnetic moment parallel to the angular momentum vector, plus
some additional terms which are very small for atoms in fields realizable in the laboratory.

So, for atoms, the dominant additional term is

HB=

e

2 mc

B~·L~=−~μ·B,~

where~μ=− 2 mce L~. This is, effectively, themagnetic momentdue to the electron’s orbital angular
momentum.

The other terms can be important if a state is spread over a region much larger than an atom. We
work the example of aplasma in a constant magnetic field. A charged particle in the plasma
has the following energy spectrum

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. This problem can be
simplified using a few different symmetry operators. We work it two different ways: in one it reduces
to the radial equation for the Hydrogen atom; in the other it reduces to the Harmonic Oscillator
equation, showing that these two problems we can solve are somehow equivalent.

1.27 Local Phase Symmetry in Quantum Mechanics and the Gauge Symmetry

metry

There is a symmetry in physics which we might call theLocal Phase Symmetryin quantum
mechanics. In this symmetry we change the phase of the (electron) wavefunction by a different
amount everywhere in spacetime. To compensate for this change,we need to also make a gauge
transformation (See section 20.3) of the electromagnetic potentials. They all must go together like
this.

ψ(~r,t) → e−i

̄hcef(~r,t)

ψ(~r,t)

A~ → A~−∇~f(~r,t)

φ → φ+

1

c

∂f(~r,t)

∂t

The local phase symmetry requires that Electromagnetism exist and have a gauge symmetry so that
we can keep the Schr ̈odinger Equation invariant under this phase transformation.