Advanced Solid State Physics

So with eqn. (67) the momentum can be calculated as

px=

∂L

∂vx

=mvx+qAx (69)

In this case, beside the kinetic momentum, there is an additional part, which is called the field
momentum. When a charged particle is moving, there is a current flowing, which causes a magnetic
field. So when the particle is decelerated, the inductance of the magnetic field keeps pushing it forward,
so that there is more energy needed to stop the particle than just the kinetic energy.
With this information the Hamiltonian can be constructed:

H=v·p−L

H=

1

2 m

(p−qA(r,t))^2 +qV(r,t)

Finally, with the substitutionp→−i~∇, the Schroedinger equation for electrons in an electromagnetic
field is obtained:

i~

∂Ψ

∂t

=

1

2 m

(−i~∇−qA(r,t))^2 Ψ +qV(r,t)Ψ (70)

9.3 Magnetic Response of Atoms and Molecules

Sometimes it is a good approximation to treat a solid as a noninteracting gas of magnetic atoms or
molecules. The Schroedinger equation reads as follows:

1

2 m

(−i~∇−qA)^2 Ψ +V(r)Ψ =EΨ

It is possible to separate the Hamiltonian into two parts:

H 0 =

−~^2

2 m

∇^2 +V(r)

H 1 =

1

2 m

(

i~∇·A+i~qA·∇+q^2 A^2

)

+gμBSzBz

with g the g-factor which is for the spin of an electron∼ 2.
H 0 is the Hamiltonian for zero magnetic field andH 1 contains the additional parts that come from the
magnetic field plus an extra term for the spin (in non-relativistic quantum mechanics one just adds
the term for the spin, there is no logical approach for that). The vector potential can be chosen so
that

∇×A=Bzˆz (this means thatBpoints into the z-direction).

∇·A= 0 (Coulomb gauge)

→A=





−^12 yBz

1

2 xBz

0



