QuantumPhysics.dvi

11 External Magnetic Field Problems

In the chapter on perturbation theory, we have studied the effect of small magnetic fields on atoms
and ions via the Zeeman effect. The application of strong magnetic fields produces further, and
remarkable effects, such as the Quantized Hall Effect. In this chapter, we shall solve (anew) the
general problem of charged particles in the presence of a strong magnetic field, and then apply the
methods and results to the case of the quantized Hall Effect.

11.1 Landau levels

In problem set 8 of 221A, we studied the problem of a 2-dimensional particle with electric charge
2 ein the presence of a uniform magnetic field, given by the following classical Lagrangian,

L(x,y,x, ̇ y ̇) =

1

2

m( ̇x^2 + ̇y^2 )−

1

2

mω^2 (x^2 +y^2 ) +

1

2

eB(xy ̇−yx ̇) (11.1)

This Lagrangian essentially also applies to a 3-dimensional charged particle in a magnetic field, since
the direction along the magnetic field will be decoupled fromthe dynamics of the other directions.
The corresponding canonical momenta are

px = mx ̇−

1

2

eBy

py = my ̇+

1

2

eBx (11.2)

The Hamiltonian is then given by

H(x,y,px,py) =

1

2 m

(px+

1

2

eBy)^2 +

1

2 m

(py−

1

2

eBx)^2 +

1

2

mω^2 (x^2 +y^2 ) (11.3)

Note that since we are not assuming the magnetic field to be small, we keep the term of orderB^2.
In problem set 8, functional methods were used to derive the partition function, and it was found
to be given by

Z= Tre−βH=

e−

(^12) β ̄hω+
1 −e−β ̄hω+

×

e−

(^12) β ̄hω−
1 −e−β ̄hω−

(11.4)

where the frequenciesω±≥0 are given by

ω±=ωB±

∣∣

∣∣eB

2 m

∣∣

∣∣ ω^2 B=ω^2 +e

(^2) B 2
4 m^2

(11.5)

The full spectrum is deduced by expanding the denominators in a Taylor series and we find,

E(n+,n−) =

1

2

̄hω+(1 + 2n+) +

1

2

̄hω−(1 + 2n−) (11.6)

forn±≥0. Effectively, the system decomposes into two independent harmonic oscillators, with
frequenciesω+andω−. Clearly, something interesting happens whenω→0, since thenω−→0,