Greg Egan on Nostr: The angular momentum of a quantum particle around any point is usually quantised, ...

The angular momentum of a quantum particle around any point is usually quantised, taking on integer multiples of ħ (the reduced Planck constant).

But in a toy universe with the topology of a torus, i.e. a square whose opposite sides are joined, as well as integer values of angular momentum, a continuum of values is possible!

For a wave function with definite angular momentum around some point, the phase (represented by the hue in the animation) undergoes an integer number of cycles as you move around the point. The value of the angular momentum is proportional to the rate at which the wave function’s phase changes with respect to the angle around the point.

But in this toy universe, if the particle is far enough from the chosen point, the total range of angles available to it is less than the usual value, 2π, by an amount that depends on the distance.

This allows the phase to complete an integer number of cycles while changing at a different rate with respect to the angle. Because there is a continuum of different total angles available, the possible values for the angular momentum are also continuous – but now the distance of the particle from the point can only take on one or more discrete values.

“Orbital angular momentum can take non-integer values in a closed universe”
Daniel Burgarth, Paolo Facchi
https://arxiv.org/abs/2506.03254v1