QuantumPhysics.dvi

and the partition function will diverge in this limit. Below, we investigate in the operator language,
the phenomenon that drives this divergence.
It is straightforward to recover this result in the operatorformulation. We setω= 0, so that
ωB=|eB/ 2 m|, and introduce the following harmonic oscillator combinations,

a 1 =

√^1

2 m ̄hωB

(ipx+mωBx)

a 2 =

1

√

2 m ̄hωB

(ipy+mωBy) (11.7)

and their Hermitian conjugates. By construction they obey canonical commutation relations
[ai,a†j] =δijfori,j= 1,2. Re-expressing the Hamiltonian in terms of these oscillators,

H= ̄hωB

(

a† 1 a 1 +a† 2 a 2 + 1

)

+i

eB ̄h

2 m

(

a† 1 a 2 −a† 2 a 1

)

(11.8)

Finally, we make the following orthonormal change of variables,

a ≡

√^1

2

(a 1 +ia 2 )

b ≡

1

√

2

(a 1 −ia 2 ) (11.9)

and their conjugates. These oscillators still obey canonical commutation relations,

[a,b] = [a,b†] = 0

[a†,b] = [a†,b†] = 0

[a,a†] = [b,b†] = 1 (11.10)

Using the following relations

2 a†a = a† 1 a 1 +a† 2 a 2 +i

(

a† 1 a 2 −a† 2 a 1

)

2 b†b = a† 1 a 1 +a† 2 a 2 −i

(

a† 1 a 2 −a† 2 a 1

)

(11.11)

the Hamiltonian may be recast as follows,

H=

1

2

̄hω+(1 + 2a†a) +

1

2

̄hω−(1 + 2b†b) (11.12)

The above spectrum then follows at once.

Asω→0, the problem is reduced to that of a charged particle in a magnetic field, and the
frequencyω−→0. Remarkably, as a result, the spectrum becomes infinitely degenerate, since the
energies do not depend uponn−any more. Alternatively, asω= 0, we have

[H,b] = [H,b†] = 0 (11.13)

For each value ofn+, there is aLandau levelwith an infinite degeneracy. The algebra ofband
b†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 quiteuniform. Nonetheless, this infinite
degeneracy makes the Landau levels an incredibly interesting phenomenon.