  Due to a special nature of the Landau problem, in which the magnetic field is
uniformly spreading over the whole two-dimensional plane, there necessarily
exist three conserved quantities, i.e. two conserved momenta and one conserved
orbital angular momentum for the electron, independently of the choice of the
gauge potential. Accordingly, the quantum eigen-functions of the Landau problem
can be obtained by diagonalizing the Landau Hamiltonian together with one of
the above three conserved operators with the result that the quantum mechanical
eigen-functions of the Landau problem can be written down for arbitrary gauge
potential. The purpose of the present paper is to clarify the meaning of gauge
choice in the Landau problem based on this gauge-potential-independent
formulation, with a particular intention of unraveling the physical
significance of the concept of gauge-invariant-extension of the canonical
orbital angular momentum advocated in recent literature on the nucleon spin
decomposition problem. At the end, our analysis is shown to disclose a
physically vacuous side face of the gauge symmetry.
