  By perceiving gauge invariance as an analytical tool in order to get insight
into the states of the "generalized Landau problem" (a charged quantum particle
moving inside a magnetic, and possibly electric field), and motivated by an
early article that correctly warns against a naive use of gauge transformation
procedures in the usual Landau problem (i.e. with the magnetic field being
static and uniform), we first show how to bypass the complications pointed out
in that article by solving the problem in full generality through gauge
transformation techniques in a more appropriate manner. Our solution provides
in simple and closed analytical forms all Landau Level-wavefunctions without
the need to specify a particular vector potential. This we do by proper
handling of the so-called pseudomomentum K (or of a quantity that we term
pseudoangular momentum Lz), a method that is crucially different from the old
warning argument, but also from standard treatments in textbooks and in
research literature (where the usual Landau wavefunctions are employed -
labeled with canonical momenta quantum numbers). Most importantly, we go
further by showing that a similar procedure can be followed in the more
difficult case of spatially-nonuniform magnetic fields: in such case we define
K and Lz as plausible generalizations of the previous ordinary case, namely as
appropriate line integrals of the inhomogeneous magnetic field - our method
providing closed analytical expressions for all stationary state
wavefunctions.. (see paper) ... also provide explicit Berry's phase
calculations and their connection to probability currents and ...issues in
elementary Quantum Mechanics and Condensed Matter Physics. As an added feature,
we also show how the possible presence of an additional electric field can be
treated through a further generalization of pseudomomenta and their proper
handling.
