\ (1,-2) \dots \cr 
1 \phantom{aaaaaaaaaaaaaaaa} 0 \phantom{aaaaaaaaaa} \phantom{(0,2) \ \ (1,1)} \ \  (0,0) \ \ (0,-1) \ \ (0,-2) \dots \cr
\end{aligned}$$

[Degeneracy of the Lowest Landau Level (LLL) Wave Function]{}
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To better understand the wave functions and issues associated with them we focus on the lowest Landau level (LLL). For the LLL, $n_r = 0$ and $\ell \leq0$. Then $E_0 = \hbar \omega_c/2$ and the associated Laguerre polynomial is identically unity. So an infinite set of degenerate LLL wave functions is given simply by $$\Psi_{LLL}(r,\phi)=\frac{e^{i\ell\phi}}{\sqrt{2\pi}}\left(\frac{r}{\sqrt{2}l_B}\right)^{|\ell|}\frac{e^{-\frac{r^2}{4l_B^2}}}{l_B\sqrt{|\ell|!}}  \ \ \ \ \ \ell \le 0.
    \label{lll_wavefunction}$$ The corresponding probability density $|\Psi_{LLL}|^2$ is a radial Gaussian function centred around $r=0$ but with a maximum at $r_{\ell}=l_B\sqrt{2|\ell|}$ with a spread about this maximum of $\sim\mathcal{O}(l_B)$. As $|\ell|$ increases, the location of the maximum of $|\Psi_{LLL}|^2$, i.e. $r_{\ell}$, moves outwards [*from zero*]{}. On the other hand, increasing the magnetic field $B$ (thus decreasing $l_B$) leads to a decrease in the position of the maximum in $|\Psi_{LLL}|^2$.

It is important to note that these results are all gauge-dependent, even given that we use the Symmetric gauge. In particular, the choice of Symmetric gauge with $A_s$ defined in the first paragraph of the 