Orbital angular momentum can be any fraction of reduced Planck constant in a coil

We use the particle-in-a-box model. We make a "coil" from a tube so that the ends of the tube are joined. The particle is inside the tube.

If the coil has just one loop, then it is the torus. The de Broglie wavelength of the particle is

      λ = h / p,

where p is the momentum of the particle.

Let the radius of the torus be r. A stationary state must have an integer number n of wavelengths along the tube:

       2 π r = n λ

                 = n h / p.

The angular momentum L of the particle around the center of the torus is

       L = p r

          = n h / (2 π)

          = n ħ.

We have the familiar quantization of angular momentum.

Let us make more loops in the coil, say N loops. Now the radius of the coil is

       r' = r / N,

and the angular momentum is

       L = p r'

           = n / N * ħ.

We have a simple example to support our claim in the previous blog post: the angular momentum of a complex system can be any fraction of ħ.

Conclusions

A superconducting current (= supercurrent) inside the lattice of a superconductor is a complex system. Many electrons, as well as lattice vibrations are involved. There is no a priori reason why the angular momentum of the system should be an integer multiple of ħ.