QuantumPhysics.dvi

7.3 Landau Levels

For a charged particle in a magnetic field without the harmonic oscillator present, we have

ω= 0 andω−= 0, and the spectrum becomes infinitely degenerate, since the energies do

not depend uponn−any more. Alternatively, asω= 0, we have

[H,a−] = [H,a†−] = 0 (7.28)

For each value ofn+, there is aLandau levelwith an infinite degeneracy. The algebra ofa−

anda†−represent the symmetry algebra of this degeneracy. Of course,in any physical system,

space is not truly of infinite extent and the magnetic field is not quite uniform. Nonetheless,

this infinite degeneracy makes the Landau levels an incredibly interesting phenomenon.

7.3.1 Complex variables

The wave functions in each Landau level exhibit remarkable properties. To exhibit these

properties, we change to complex variables. Thex,yparts of the combinationsa±correspond

to forming the complex variables

z=

1

√

2

(x+iy) pz=

1

√

2

(px−ipy)

̄z=

1

√

2

(x−iy) p ̄z=

1

√

2

(px+ipy) (7.29)

which satisfy the canonical commutation relations,

[z,pz] = [ ̄z,pz ̄] = i ̄h

[z,pz ̄] = [ ̄z,pz] = 0 (7.30)

and as a result,

pz=−i ̄h

∂

∂z

p ̄z=−i ̄h

∂

∂z ̄

(7.31)

The oscillators now become,

a+=

1

√

2 m ̄hωB

(

ip ̄z+mωBz

)

a†+=

1

√

2 m ̄hωB

(

−ipz+mωBz ̄

)

a−=

1

√

2 m ̄hωB

(

ipz+mωB ̄z

)

a†−=

1

√

2 m ̄hωB

(

−ip ̄z+mωBz

)

(7.32)

Sinceω= 0, the Hamiltonian depends only ona+anda†+,H= ̄hωB(2a†+a++1). The lowest

Landau level| 0 ,n−〉is characterized by a+| 0 ,n−〉= 0. The wave functions of the lowest