aspects of the problem with a focus on accidental symmetries. This section is a preparation for the case of a finite periodic volume to be discussed in the next section. In the following, we will use natural units in which $\hbar = c = 1$.

Classical Treatment
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Ignoring its spin, we consider a non-relativistic electron of mass $M$ and electric charge $-e$ moving in a constant magnetic field $\vec B = B \vec e_z$, which we realize through the vector potential $$A_x(\vec x) = 0, \ A_y(\vec x) = B x, \ A_z(\vec x) = 0.$$ Since the motion along the direction of the magnetic field is trivial, we restrict ourselves to 2-dimensional motion in the $x$-$y$-plane. Obviously, this is just standard cyclotron motion. To get started, in this subsection we treat the problem classically. The particle then experiences the Lorentz force $$\vec F(t) = - e \vec v(t) \times B \vec e_z,$$ which forces the particle on a circular orbit of some radius $r$. It moves along the circle with an angular velocity $\omega$, which implies the linear velocity $v = \omega r$ and the acceleration $a = \omega^2 r$. Hence, Newton’s equation takes the form $$m \omega^2 