result, the quantized Landau levels have a continuous infinite degeneracy. The energy eigenstates are shifted one-dimensional harmonic oscillator wave functions $\psi_n(x)$, i.e. $$\label{eigenstates}
\langle \vec x|n p_y \rangle = \psi_n\left(x + \frac{p_y}{M \omega} \right)
\exp(i p_y y).$$ Similarly, one can construct eigenstates of the generator $P_x = - i {\partial}_x + e B y$ of infinitesimal translations (up to a gauge transformation) in the $x$-direction $$\label{eigenx}
\langle \vec x|n p_x \rangle = \psi_n\left(y - \frac{p_x}{M \omega} \right)
\exp(i p_x x) \exp(- i e B x y).$$ It is straightforward to show that the two sets of eigenstates $\langle \vec x|n p_y\rangle$ and $\langle \vec x|n p_x\rangle$ span the same subspace of localized states in the Hilbert space.

Since all classical orbits are closed and the center of the cyclotron orbit plays the role of a Runge-Lenz vector, it is natural to ask whether the degeneracy is caused by an accidental symmetry. Of course, since the Runge-Lenz vector plays a dual role and is also generating translations (up to gauge transformations), in this case the “accidental” symmetry would just be translation invariance. Indeed, in complete analogy to the classical case, it is easy to convince oneself that $[H,R_x] = [H,R_y] = 