11.2 Complex variables
Thex,yparts of the combinationsaandbcorrespond 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) (11.14)which satisfy the canonical commutation relations,
[z,pz] = [ ̄z,p ̄z] = i ̄h
[z,p ̄z] = [ ̄z,pz] = 0 (11.15)and as a result,
pz=−i ̄h∂
∂z
p ̄z=−i ̄h∂
∂z ̄(11.16)
The oscillators now become,
a=1
√
2 m ̄hωB(
ip ̄z+mωBz)
a†=1
√
2 m ̄hωB(
−ipz+mωBz ̄)b=1
√
2 m ̄hωB(
ipz+mωB ̄z)
b†=1
√
2 m ̄hωB(
−ipz ̄+mωBz)
(11.17)Since we are assumingω= 0, the Hamiltonian depends only onaanda†,H= ̄hωB(2a†a+ 1).
The lowest Landau level| 0 ,n−〉is characterized bya| 0 ,n−〉= 0. The wave functions of the lowest
Landau level satisfy the differential equation
(
∂
∂z ̄
+
mωB
̄hz)
ψ(z,z ̄) = 0 (11.18)Its general solution is straightforward (since as a differential equation in ̄z, you may think ofzas
a constant coefficient), and we get
ψ(z, ̄z) =φ(z)ψ(0)(z,z ̄) ψ(0)(z,z ̄) = exp{
−mωB
̄h
|z|^2}
(11.19)whereφ(z) is an arbitrary complex analytic function ofz. Actually,φ(z) cannot has poles since they
would lead to non-square integrable wave functions. Also, assuming thatψ(z,z ̄) is single-valued,
it follows thatφ(z) must be single-valued, and thus cannot have branch cuts. Therefore,φ(z)
must be single-valued and holomorphic throughout the complex plane. A basis for such functions
is obtained by polynomials (and square integrable Taylor expandable functions by completeness).
Notice that all such polynomial functions are obtained by applyingb†repeatedly to the ground
stateψ(0)of bothaandb, since we have
ψn(0)−∼(b†)n−ψ 0 (z, ̄z)∼zn−ψ 0 (z,z ̄) (11.20)This agrees with general principles that the group theory ofb,b†must precisely reproduce the
normalizable spectrum.