  One intriguing issue in the nucleon spin decomposition problem is the
existence of two types of decompositions, which are representably characterized
by two different orbital angular momenta (OAMs) of quarks. The one is the
manifestly gauge-invariant mechanical OAM, while the other is the so-called
gauge-invariant canonical (g.i.c.) OAM, the concept of which was introduced by
Chen et al. To get a deep insight into the difference of these two
decompositions, it is therefore vitally important to understand the the
physical meanings of the above two OAMs correctly. Also to be clarified is the
implication of the gauge symmetry that is immanent in the concept of g.i.c.
OAM. We find that the famous Landau problem provides us with an ideal tool to
answer these questions owing to its analytically solvable nature. After
deriving a complete relation between the standard eigen-functions of the Landau
Hamiltonian in the Landau gauge and in the symmetric gauge, we try to unravel
the physics of the the canonical OAM and the mechanical OAM, by paying special
attention to their gauge-dependence. We also argue that, different from the
mechanical OAM of the electron, the canonical OAM or its gauge-invariant
version would not correspond to any direct observables at least in the Landau
problem. Also briefly discussed is the uniqueness or non-uniqueness problem of
the nucleon spin decomposition, which arises from the arbitrariness in the
definition of the so-called physical component of the gauge field.
