Figure 62: Left: Fermi sphere for zero applied magnetic field. Right: Split of Fermi sphere into
different Landau levels when magnetic field is applied.
This is the equation for a harmonic oscillator
−~^2
2 m∂^2 g(x)
∂x^2+
1
2
mω^2 (x−x 0 )^2 g(x) =Eg(x)Ekz,n=~^2 k^2 z
2 m
+~ωc(
n+1
2
)ωc=
qBz
m
The equation holds for a magnetic field in the z-direction because then the electrons can have kinetic
energy in the z-direction without being pushed in this direction. So the oscillation just takes place in
the x-y-plane, where the electrons move in a circle with the cyclotron frequency. Because of this, the
value in kinetic energy in the z-direction is not restricted, but only discrete values in the x-y-plane
are allowed (going with n), because therek^2 x+ky^2 has to be a constant when the electrons move in a
circle.
So the harmonic motion is quantized and n labels the Landau level.
En=~ωc(
n+1
2
)
=~^2
2 m(kx^2 +k^2 y)Fig. 62 shows the splitting of the fermi sphere in cylinders, corresponding to the different Landau
levels. Note that the cylinders have a certain height because the energy in z-direction gets smaller
when the energy in the x-y plane gets bigger. Of course, there would be states at cylinders even with
infinite height, but those would not be occupied. Here only occupied states are depicted and we know
that they are within a fermi-sphere.