\right) \\[16pt]
    \label{kin_cent_pot2}$$ allows one to easily show that the minimum occurs at $x = 1$ ($r = r_\ell$) and the wave function (not surprisingly) is maximized where the effective potential is a minimum, [*independent of the sign of $\ell$*]{}. The value of the minimum potential is zero for all $\ell \le 0$ (independent of the magnitude of $\ell$), while it is $\hbar \omega_c |\ell|$ for $\ell > 0$ (grows linearly with $\ell$). In fact the difference between positive and negative $\ell$ is more general than this, and one can write that for [*all*]{} $r$, $$V_{|\ell|}(r) = V_{-|\ell|}(r) + \hbar\omega_c |\ell|,
  \label{pot_shift}$$ so there is a constant shift in potential values. The constant shift is consistent with attaining the same probability density for different energies.

![ $V_\ell(r)/(\hbar \omega_c)$ vs. $r/\ell_B$ for various values of $\ell$, both negative and positive. Solid lines correspond to negative $\ell$ while dashed lines correspond to positive $\ell$, and solid and dashed lines of the same colour correspond to the same magnitude, $|\ell|$ and hence have the same $r_\ell$ at which their minima occur. As explained in the text, for negative $\ell$ the minimum value of $V_\ell$ is always at zero while for positive $\ell$, the 