QuantumPhysics.dvi

and their adjointsa† 1 anda† 2. By construction they obey canonical commutation relations

[ai,a†j] =δij fori,j= 1,2, while [a 1 ,a 2 ] = [a† 1 ,a† 2 ] = 0. Re-expressing the Hamiltonian in

terms of these oscillators, we find,

H= ̄hωB

(

a† 1 a 1 +a† 2 a 2 + 1

)

+i

eB ̄h

2 m

(

a† 1 a 2 −a† 2 a 1

)

(7.19)

Finally, we make the following orthonormal change of variables froma 1 , 2 toa±,

a±≡

1

√

2

(a 1 ±ia 2 ) (7.20)

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

[ai,aj] = [a†i,a†j] = 0 [ai,a†j] =δi,j i,j= +,− (7.21)

Using the following relations (the±signs are correlated),

2 a†±a±=a† 1 a 1 +a† 2 a 2 ±i

(

a† 1 a 2 −a† 2 a 1

)

(7.22)

the Hamiltonian may be recast as follows,

H=

1

2

hω ̄ +

(

1 + 2a†+a+

)

+

1

2

̄hω−

(

1 + 2a†−a−

)

(7.23)

where the combinationsω±are defined by

ω±=ωB±

∣∣

∣∣eB

2 m

∣∣

∣∣≥ 0 (7.24)

We have now achieved our goal: the magnetic field Hamiltonian has beenexpressed in terms

of two independent harmonic oscillator variablesa,b.

To solve for the spectrum is straightforward. The ground state| 0 , 0 〉satisfiesa±| 0 , 0 〉= 0,

and the excited states are given by

|n+,n−〉=N(n+,n−) (a+)n+(a−)n−| 0 , 0 〉 n±≥ 0 (7.25)

whereNis the normalization factor. The corresponding eigenvalue is simply,

E(n+,n−) =

1

2

̄hω+(1 + 2n+) +

1

2

̄hω−(1 + 2n−) (7.26)

Two different states|n+,n−〉and|n′+,n′−〉will be degenerate provided

ω+(n′+−n+) +ω−(n′−−n−) = 0 (7.27)

Sincen±,n′±are integers, this relation can have solutions if and only ifω−/ω+isrational

(including the value 0). If this is the case, we writeω±=k±ω 0 wherek+>0 andk−≥ 0

are relatively prime integers. All states degenerate with|n+,n−〉are then given by|n++

nk−,n−−nk+〉for any integer nsuch thatn− ≥ nk+. Ifω−= 0, namelyω = 0, this

degeneracy is infinite, asncan take an infinite number of different values, while ifω−>0,

the degeneracy is necessarily finite.