wang
(Wang)
#1
discovery for the scattering Aharonov-Bohm effect, that the presence of an integer number of
basic flux quanta has no measurable effects. For a half flux quantumν= 1/2, the particle is
quantized as a spin 1/2 fermion! Denoting byE 0 the ground state energy forℓ+ν= 0, then
adding angular momentum will increase the energy. In figure 9, we show a qualitative picture
of how the ground state energy depends onν. Note that for|ν|< 1 /2, the ground state has
ℓ= 0, but asνis increased, the ground state is achieved successively forℓ=− 1 ,ℓ=− 2 ,···.
Figure 9: Energy of the ground state for the bound state Aharonov-Bohm effect
Finally, for completeness, we obtain the full spectrum using Besselfunctions. The two
linearly independent solutions of (7.48) are the Bessel functionsJ|ℓ+ν|(kr) andN|ℓ+ν|(kr),
with smallr asymptotics given respectively byr|ℓ+ν|andr−|ℓ+ν|. The general solution to
(7.48) is then given by,
ψℓ(r) =αJ|ℓ+ν|(kr) +βN|ℓ+ν|(kr) (7.50)
The boundary conditions impose a quantization condition onk,
J|ℓ+ν|(kR+)N|ℓ+ν|(kR−)−J|ℓ+ν|(kR−)N|ℓ+ν|(kR+) = 0 (7.51)
In the limit where the inner cylinder has small radius,R−→0, we may neglect the sec-
ond term in this equation, sinceJ|ℓ+ν|(kR−)→0 then, and we are left with the condition
J|ℓ+ν|(kR+) = 0, so thatkR+is given by the zeros of the Bessel functionJ|ℓ+ν|. Analysis with
Maple, for example, readily confirms numerically that the corresponding energies indeed do
depend uponν.
7.5 The Dirac magnetic monopole
The existence of a fundamental quantum of magnetic flux has an immediate application to
the theoretical existence ofmagnetic monopolesin quantum mechanics. One of Maxwell’s
equations,∇·~ B= 0, states that, classically, there exist no magnetic monopoles. The electric
counterpart of this equation is∇·~ E=ρ, whereρis the electric charge density, which can
