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

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