QuantumPhysics.dvi

11.3 Calculation of the density of states in each Landau level

An important characterization of the Landau levels is the number of quantum states that are
available. This is easy to compute as we have the explicit wave-functions available. Consider the
lowest Landau level first. All states in this level are obtained by applying powers ofb†to the ground
state with wave functionψ 0 (z,z ̄). The result is (the overall normalization will be immaterial),

ψ(0)n (z,z ̄)∼znexp

{

−

∣∣

∣∣eB

2 ̄h

∣∣

∣∣×|z|^2

}

(11.21)

In terms of radial coordinates,x+iy=r eiθ, we have

ψ(0)n (r,θ)∼rneinθ×exp

{

−

∣∣

∣∣eB

4 ̄h

∣∣

∣∣×r^2

}

(11.22)

Clearly, because of theθ-dependence, states for differentnare orthogonal to one another. The
maximum of the probability densityψ(0)n (r,θ) is attained at radius

r^2 n= 2n

∣∣

∣∣ ̄h

eB

∣∣

∣∣ (11.23)

But this means that a given areaπR^2 ≫| ̄h/eB|will contain a number of statesNgiven by

N=πR^2

∣∣

∣∣eB

2 π ̄h

∣∣

∣∣ (11.24)

Hence the density of states of the first Landau level for magnetic fieldB >0 is given by

nB=

N

πR^2

=

eB

2 π ̄h

(11.25)

This is a classic result, you may find in Landau and Lifschitz. In a higher Landau level, the wave
function will have an extra factor of ̄zn+. But for given, finiten+, this extra factor will not modify
the asymptotic behavior of the density of states found in thelowest Landau level, so that the
formula (11.25) is valid for any Landau level.

11.4 The classical Hall effect

The set-up of the Hall effect (classical or quantum) is a conductor subject to an electric fieldE
and a perpendicular magnetic fieldB. For zero magnetic field, the electric field creates an electric
currentjwhich is parallel toE. The zero magnetic field conductivity is denotesσ 0 and relates
j=σ 0 E. If the current is produced by individual charge carriers such as electrons, then the current
is given by the density of carriersnand their electric chargeebyj=nev.

When the magnetic field is turned on, charge carriers are deviated and produce an electric
current, orHall current, transverse to the electric field. The strength of this current may be
obtained from the Lorentz force formula,

me

dv

dt

=eE+ev×B−

ne^2

meσ 0

v (11.26)