  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 {\it gauge-variant} quantity, and observation of a
gauge-variant quantity would contradict a fundamental principle of physics
called the {\it gauge principle}. In recent researches, however, Bliohk 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 {\it particle-like} aspect but also to the
{\it wave-like} aspect of the Landau electron.
