Advanced Solid State Physics

The conjugated variable to position has now two components, the first therm in eqn.(9)mvwhich is
the normal momentum. The second termqAwhich is called the field momentum. It enters in this
equation because when a charged particle is accelerated it also creates a Magnetic field. The creation
of the magnetic field takes some energy, which is then conserved in the field. When one then tries to
stop the moving particle one has to overcome the kinetic energy plus the energy that is conserved in
the magnetic field, because the magnetic field will keep pushing the charged particle in the direction
it was going ( this is when you get back the energy that went into creating the field, also called self
inductance).

To construct the Hamiltonian we have to perform a Legendre transformation from the velocityvto
the generalized momentump.

H=

∑^3

i=

pivi(pi)−L(r,v(p,t)

H=p·

1

m

(p−qA)−

1

2 m

(p−qA)^2 +qφ−

q

m

(p−qA)·A

Which then reduces to

H=

1

2 m

(p−qA)^2 +qφ (10)

Eqn.(10) is the Hamiltonian for a charged particle in an EM-field without Spin. The QM Hamiltonian
of a single Spin in a magnetic field is given by

H=−mˆ·B (11)

where

mˆ =

2 gμB

~

Sˆ (12)

μB = 2 qm~ the Bohr Magnetron andSˆ is the QM Spin operator. By Replacing the momentum and
location in the Hamiltonian by their respective QM operators we obtain the QM Hamiltonian for our
System

Hˆ=^1

2 m

(pˆ−qA(ˆr))^2 +qφ(ˆr)−

2 gμB

~

Sˆ·B(ˆr) (13)

We consider the simple case of constant magnetic field pointing in the z-directionB= (0, 0 ,Bz)and
φ= 0. Plugging this simplifications in eqn.(13) yields

Hˆ=^1

2 m

(pˆ−qA(ˆr))^2 −

2 gμB

~

BzSˆz≡Hˆ 1 +Sˆ (14)

We see that the Hamiltonian consists of two non interacting partsHˆ 1 = 21 m(pˆ−qA(ˆr))^2 andSˆ=
−^2 gμ~BBzSˆz. The state vector of an electron consists of a spacial part and a Spin part