Magnetic monopole and quantisation of angular momentum

Question

I read chapter 6.12/6.13 in J. D. Jackson's Classical Electrodynamics about the magnetic monopole and a certain detail is confusing me.

First in a semiclassical consideration of a magnetic monopole, the fact that change of angular momentum must occur in integral multiples of $\hbar$ is used to show that magnetic and electric charge must have discrete values.

Then a simplified discussion is given of Dirac's original argument that leads to the same quantisation condition for electric and magnetic charge. But in this presentation - when I understand it correctly - the quantisation of angular momentum is not used. Instead, single-valuedness of wave functions is used, together with gauge invariance.

So I am not sure, whether quantisation of angular momentum is really needed to find the quantisation condition?

Of course when not, the quantisation condition for electric/magnetic charge would explain the quantisation of angular momentum, wouldn't it?

Answer 1

1. Well, the point is that the quantization of angular momentum arises semiclassically from the WKB approximation by imposing that the wave function should be single-valued. Here we use an azimuthal angle and its corresponding angular momentum $(\varphi,L_z)$ as canonical variables for simplicity rather than the 3D position and momentum, cf. the geometric setup mentioned in Jackson. The single-valuedness condition becomes a periodicity condition $$ \psi(\varphi+2\pi)~=~\psi(\varphi).$$ This leads to the Bohr-Sommerfeld quantization rule $$ \oint \!L_z ~\mathrm{d}\varphi~\in~h\mathbb{Z} ,$$ which in turn leads to the quantization condition for the angular momentum $L_z$.

2. Hence both (i) the above semiclassical argument for angular momentum quantization and (ii) Dirac's magnetic monopole quantization argument rely on the single-valuedness of the wavefunction.