QuantumPhysics.dvi

Figure 8: Aharonov-Bohm bound state set-up

Lz=−i ̄h∂/∂θcommutes with the Hamiltonian, and may be diagonalized simultaneously.

Thus, we are interested in wave functions for fixed angular momentumℓ,

ψ(r) =ψℓ(r)eiℓθ (7.45)

The covariant derivative acts as follows,

−i ̄h∇ψ(r)−eA(r)ψ(r) =−i ̄heiℓθ

(

nr

∂

∂r

+

nθ

r

(iℓ+iν)

)

ψℓ(r) (7.46)

where we have defined the combination

ν=

eΦB

2 π ̄h

(7.47)

The radial part of the Schr ̈odinger equation becomes,

r^2 ψ′′ℓ+rψ′ℓ+

(

k^2 r^2 −(ℓ+ν)^2

)

ψℓ= 0 (7.48)

The infinite potentialV outsideR−< r < R+imposes the following boundary conditions,

ψℓ(R+) =ψℓ(R−) = 0 (7.49)

The differential equation is of the Bessel kind, and can be easily solved in terms of those

functions. Before we do so, however, much can already be deduced from inspection of (7.48).

The most striking property is that the equation depends only on thecombinationℓ+ν, not

onℓandνseparately. In particular, the effect of adding a single flux quantum,ν= 1, simply

has the effect of moving the spectrum up by one unit of angular momentum,ℓ→ℓ+ 1, but

leaving the entire spectrum (for all angular momenta) unchanged.This confirms our earlier