  There is an interesting but not so popular quantity called pseudo orbital
angular momentum (OAM) in the Landau-level system, besides the well-known
canonical and mechanical OAMs. The pseudo OAM can be regarded as a
gauge-invariant extension of the canonical OAM, which is formally gauge
invariant and reduces to the canonical OAM in a certain gauge. Since both of
the pseudo OAM and the mechanical OAM are gauge invariant, it is impossible to
judge which of those is superior to the other solely from the gauge principle.
However, these two OAMs have totally different physical meanings. The
mechanical OAM shows manifest observability and clear correspondence with the
classical OAM of the cyclotron motion. On the other hand, we demonstrate that
the standard canonical OAM as well as the pseudo OAM in the Landau problem are
the concepts which crucially depend on the choice of the origin of the
coordinate system. We try to reveal the relation between the pseudo OAM and the
mechanical OAM as well as their observability by paying special attention to
the role of guiding-center operator in the Landau problem.
