Revisiting the compatibility problem between the gauge principle and the observability of the canonical orbital angular momentum in the Landau problem

Abstract

As is widely-known, the eigen-functions of the Landau problem in the symmetric gauge are specified by two quantum numbers. The first is the familiar Landau quantum number n, whereas the second is the magnetic quantum number m, which is the eigen-value of the canonical orbital angular momentum (OAM) operator of the electron. The eigen-energies of the system depend only on the first quantum number n, and the second quantum number m does not correspond to any direct observables. This seems natural since the canonical OAM is generally believed to be a gauge-variant quantity, and observation of a gauge-variant quantity would contradict a fundamental principle of physics called the gauge principle. In recent researches, however, Bliokh et al. analyzed the motion of helical electron beam along the direction of a uniform magnetic field, which was mostly neglected in past analyses of the Landau states. Their analyses revealed highly non-trivial m-dependent rotational dynamics of the Landau electron, but the problem is that their papers give an impression that the quantum number m in the Landau eigen-states corresponds to a genuine observable. This compatibility problem between the gauge principle and the observability of the quantum number m in the Landau eigen-states was attacked in our previous letter paper. In the present paper, we try to give more convincing answer to this delicate problem of physics, especially by paying attention not only to the particle-like aspect but also to the wave-like aspect of the Landau electron.