Landau levels in 2D I seem to be having a very basic misunderstanding of Landau levels in 2D. Consider the derivation give on page 7 here, where we consider electrons free to move in 2 dimensions, with a magnetic field applied perpendicular to the plane. The wave function they get is 
$\psi(x,y)=\exp(ik_y y)u_n(x-x_0)$
In particular, if you square this to get the probability density, you get 
$|\psi|^2(x,y)=u_n^2(x-x_0)$ which does not depend on $y$. This makes no sense at all to me. Any sort of illustration I can find of this shows localized electron orbits around points in the x-y plane, analogous to the orbits of classical charged particles in a magnetic field. 
How can the probability density possibly be independent of $y$?
 A: The answer is hinted at in the bottom 3 lines of Slide 5 in your 2nd ref., http://uw.physics.wisc.edu/~himpsel/551/Lectures/Landau.pdf:

"The same B-field can be created by other vector potentials, such as $A = \frac{1}{2} (-B_z y , B_z x , 0)$. This ambiguity is called gauge symmetry."

In other words, Landau's gauge for the magnetic field is most effective for deriving a simple separable solution, but not the most relevant one for the problem's symmetry to rotations around the $z$-axis.
In Landau's gauge the solution to the localization problem is to note that the harmonic energy levels are strongly degenerate due to their dependence on $k_y$. This degeneracy allows, in principle, the construction of states localized along both $x$ and $y$. As a side note, the same kind of result would be obtained on interchanging the $x$ and $y$ directions. We'd get degenerate plane waves along $x$ and a harmonic oscillator along $y$, and we'd have to construct states localized along $x$. 
But a more elegant approach is to observe that we can choose a symmetric gauge from the beginning, reading, as pointed out in the quote, $A = \frac{1}{2} (-B_z y , B_z x , 0)$. In this case the solution is somewhat more complicated to arrive at, but produces the expected explicitly symmetric eigenstates, with harmonic oscillator energy eigenvalues and a degeneracy indexed this time by the system's angular momentum along $z$. 
See an outline on the Wikipedia page, or in these notes, or in polar coordinates here, etc.
