  The eigen-states of the Landau Hamiltonian in the symmetric gauge are
characterized by two integers $n$ and $m$. Here, $n$ denotes the familiar
Landau quantum number, while $m$ represents the eigen-value of the canonical
orbital angular momentum (OAM) operator $\hat{L}^{can}_z$. On the other hand,
the eigen-states in the 1st Landau gauge are characterized by two integers $n$
and $k_x$, here $n$ is the Landau quantum number, while $k_x$ is the
eigen-value of the canonical momentum operator $\hat{p}^{can}_x$. Since the
canonical momentum and the canonical OAM are both gauge-variant quantities,
their eigenvalues $k_x$ and $m$ are standardly believed not to correspond to
observables. However, this wide-spread view was suspected in a recent paper
based on the logical development of the gauge-potential-independent formulation
of the Landau problem, which predicts the existence of two conserved momenta
$\hat{p}^{cons}_x$ and $\hat{p}^{cons}_y$ and one conserved OAM
$\hat{L}^{cons}_z$. They are regarded as Noether charges of the Landau
Hamiltonian, the conservation of which is guaranteed {\it independently} of the
choice of the {\it auge potential}. In particular, on the basis of novel
covariant gauge transformation properties of these conserved operators, the
eigen-values of which are characterized by the quantum numbers $k_x$, $k_y$,
and $m$, it was claimed that these quantum numbers correspond to observables at
least in principle. The purpose of the present paper is to show that this claim
is not justified, regardless of the differences in the two theoretical
formulations of the Landau problem, i.e. the traditional formulation and the
gauge-potential-independent formulation.
