Probability current for electron in uniform magnetic field: wave function forever splitting apart? In this document
http://hitoshi.berkeley.edu/221a/landau.pdf
on Landau levels, in section 4, page 19,  "Transitionally
invariant Gauge", they analyze the free electron in a uniform
magnetic field problem, with the gauge $\vec{A}$ = $(0, Bx)$.
This leads to the wave function being a combination of
harmonic oscillator in the $x$ dimension and a infinite strip
in the $y$ direction, as shown in their Figure 6, page 20,
with the center of the harmonic oscillator being bumped over
by some amount which depends on the $y$-momentum(wave number)
$k_y$.
On page 21
the probability current is said to be going up on one side of
the harmonic oscillator   (where $x > \dfrac{\hbar c}{eB} k_y$)
and down on
the other.  By up and down I mean $+y$ direction and $-y$ direction.
How can a wave function have its right side be sending off
probability, forever, upwards, and left side be sending off
probability, forever, downwards?  It would seem like this is
saying as time progresses, the wave function eventually is
splitting in two with its right side going off to $+y$ infinity,
and its left side going off to $-y$ infinity, and after enough
time passes its right and left sides will have separated their
separate ways each arriving at the $+y$ vertical infinity location
in the universe and the other at the $-y$ vertical infinity..
 A: The state is an energy eigenstate and all the phase change is in the y direction in a periodic fashion, so there is an actual period $T$ where after time $T$ the wave function is exactly the same. So it isn't really changing in time, it is just changing its overall phase.
The wave in the paper is unnormalized in the y direction. And it was chosen to be a plane wave. A plane wave has a regular phase change, and hence a steady canonical momentum.
But the kinetic momentum differs from the canonical momentum by the vector potential. So the kinetic momentum is different in different places. And it is determined because we chose a phase and a gauge. You could add a constant to the vector potential and the kinetic momentum would be different, because you are fixing the canonical momentum to be nice because that made the wave mathematically easy to describe.
Since the wave is unnormalized in the y direction it extends forever in the $\pm \hat y$ directions and so isn't really changing. And we just gave it that current out of convenience. There are lots of currents consistent with the same probability. There are also lots of gauges, but normally if you adjust the gauge you also have to adjust the phase if you want the physics to be the same, but in this case we wanted the phase to be a nice plane wave.
If you look at the example where they add an electric field, effectively it is just adding a constant to the vector potential. So never read too much into a wave function, and know which parts relate to observations.
In this case nothing is really changing in time, since the global overall phase is unphysical. And we selected a certain canonical momentum from the plane wave phase and then selected a gauge and that is what made kinetic momentum be what it is. It isn't anything deeper than that.
