momentum. Still, it is important to note that $R_x$, $R_y$, and $L$ do not form an $SU(2)$ algebra. This follows because, in analogy to the classical case $$[R_x,R_y] = \frac{i}{e B},$$ i.e. $R_x$ and $R_y$ are generators of a Heisenberg algebra. As a consequence one obtains $$[R_+,R_-] = \frac{2}{e B}.$$

Creation and Annihilation Operators
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Since the particle in the magnetic field leads to the spectrum of a 1-dimensional harmonic oscillator (however, with infinite degeneracy), it is natural to ask how one can construct corresponding creation and annihilation operators such that $$H = \omega \left(a^\dagger a + \frac{1}{2}\right), \ [a,a^\dagger] = 1.$$ Remarkably, the creation and annihilation operators are closely related to the Runge-Lenz vector, i.e. the vector that points to the center of the classical cyclotron orbit. Since we have seen that $$H = \frac{1}{2} M \omega^2 r^2 = 
\frac{1}{2} M \omega^2 \left[(x - R_x)^2 + (y - R_y)^2\right],$$ one is led to identify $$a = \sqrt{\frac{M \omega}{2}} \left[x - R_x - i (y - R_y)\right], \
a^\dagger = \sqrt{\frac{M \omega}{2}} \left[x - R_x + i (y - R_y)\right],$$ which indeed have the desired properties. One also finds that $$[L,a] = - a, \ [L,a^\dagger] = a^\dagger,$$ which implies 