increasingly overlap with one another as $|\ell |$ increases.

[The Kinetic Centrifugal Potential]{}
-------------------------------------

Returning to the free electron case (still in the Symmetric gauge) note that the Landau energies clearly depend on the sign of $\ell$, and yet the wave functions (\[psi\_rad\]) clearly depend only on their absolute value, $|\ell|$. The situation where two identical probability densities correspond to completely different energies is not as strange as one might initially think; for example, the difference between a stationary vs. travelling wave packet is a simple phase factor and the energies of these two configurations differ by an arbitrary amount, as the phase factor determines the average velocity of the wave packet, and yet the probability distributions are identical in the two cases.

Here, two eigenstates with oppositely signed $\ell$ values ($|\ell|$ and $-|\ell|$), but with the same value of $n_r$, have energies $E_{n_r,|\ell|}=\left(n_r+|\ell|+\frac{1}{2}\right)\hbar\omega_c$ and $E_{n_r,-|\ell|}=\left(n_r+\frac{1}{2}\right)\hbar\omega_c$, respectively. Yet $|\Psi_{n_r,|\ell|}|^2=|\Psi_{n_r,-|\ell|}|^2$. To understand why, note that the effective potential, given by the kinetic centrifugal potential (\[ham\_symm3\]) is $$V_{\ell}(r)=\frac{\Lambda_z^2(r)}{2mr^2}=\frac{\left(\hbar \ell+\frac{1}{2}m\omega_cr^2\right)^2}{2mr^2},\\[5pt]
    \label{kin_cent_pot}$$ where $L_z$ in the original definition of $\Lambda_z(r)$, Eq. (\[kin\_ang\_mom2\]), is replaced by its eigenvalue $\hbar \ell$. Rewriting this as ($x \equiv r/r_{\ell}$)\
$$V_{\ell}(r)=\frac{\hbar \omega_c |\ell|}{4} \left( x^2 + 2\frac{\ell}{|\ell|} + \frac{1}{x^2} 