as well, We have introduced the classical cyclotron frequency $\omega_c \equiv {eB}/{m}$. Converting to polar coordinates, we get $$H=-\frac{\hbar^2}{2m}(\partial_r^2+\frac{1}{r}\partial_r)+\frac{L_z^2}{2mr^2}+\frac{\omega_c}{2}L_z+\frac{1}{8}m\omega_c^2r^2,
\label{ham_symm2}$$ where $L_z= -i\hbar (x\partial_y - y \partial_x) = -i\hbar \partial_{\phi}$ is the standard operator for the $z$ component of the angular momentum. Note that this operator is *gauge-dependent* and its eigenvalues should not be considered to represent the real, physical angular momentum of the electrons in this system. To consider the *true* angular momentum of the system, we must take into account the effect of the magnetic field. Just like the canonical momentum is shifted to the kinetic momentum in the presence of a magnetic field, we can also define a kinetic angular momentum $\mathbf{\Lambda}$, $$\mathbf{L}=\mathbf{r}\cross\mathbf{p}\rightarrow\mathbf{\Lambda}=\mathbf{r}\cross\boldsymbol{\pi}=\mathbf{r}\cross(\mathbf{p}+e\mathbf{A})=\mathbf{L}+\mathbf{r}\cross e\mathbf{A}.
    \label{kin_ang_mom}$$ Since we are working in 2 dimensions, the kinetic angular momentum is perpendicular to the plane and is given by $$\mathbf{\Lambda}(\mathbf{r})=\left(L_{z}+\frac{1}{2}m\omega_cr^2\right)\boldsymbol{\hat{z}}=\Lambda_z(r)\boldsymbol{\hat{z}},
      \label{kin_ang_mom2}$$ where $\Lambda_z(r)$ is the magnitude of the kinetic angular momentum in the $\boldsymbol{\hat{z}}$ direction. We can also rewrite Eq. (\[ham\_symm2\]) in terms of $\Lambda_z(r)$, $$H=-\frac{\hbar^2}{2m}(\partial_r^2+\frac{1}{r}\partial_r)+\frac{\Lambda_z^2(r)}{2mr^2}.
       \label{ham_symm3}$$ Analogously, this new term can be interpreted as a kinetic centrifugal potential $V_{l}(r)$.

To calculate the eigenstates and eigenvalues of Eq. (\[ham\_symm2\]), we first note that since $[H,L_z]=0$, we can write the eigenstates $\Psi(r,\phi)$ 