Landau levels degeneracy in symmetric gauge I'm reading David Tong's lecture notes on the Quantum Hall Effect.
When symmetric gauge taken, a basis of the lowest landau level wave functions is
$$\psi_{LLL,m}\sim\left(\frac{z}{l_B}\right)^m e^{-|z|^2/4l_B^2},$$
where $z=x-iy$,
and we have
$$J_z\psi_{LLL,m}=\hbar m \psi_{LLL,m}.$$
On page 25, it says that

the profiles of the wavefunctions form concentric rings around the origin. The higher the angular momentum $m$, the further out the ring.


The wavefunction with angular momentum $m$ is peaked on a ring of radius $r=\sqrt{2m}l_B$. This means that in a disc shaped region of area $A=\pi R^2$, the number of states is roughly (the integer part of)
$$N=R^2/2l_B^2=A/2\pi l_B^2$$

I can't understand  these two statements. I think the profile of $e^{-|z|^2/4l_B^2}$ does form concentric rings around the origin, but does not when multiplied by $(\frac{z}{l_B})^m$. And why $r_{max}=\sqrt{2m}l_B$?
For the second statement, my understanding is that it divide the area in real space by the area "a wave function occupies", but if this is the case, shouldn’t there be a $m$ in the denominator?
 A: *

*$r_\text{max}$ is the location where $|\psi|^2$ is maximized. Even after multiplying the wavefunction by $z^m$, $|\psi|^2$ is still symmetric under rotation around the origin (only a function of $|z|$), so in this sense the wavefunctions still represents concentric rings.


*Remember the integer $m$ actually labels different eigenstates. The area of the annulus between two neighboring states is $\pi r_{m+1}^2-\pi r_m^2=2\pi  l_B^2$, so the number of states is roughly $\frac{A}{2\pi l_B^2}$. Alternatively, as $r_{\text{max}}$ grows with $m$, the maximal value of $m$ before the wavefunction completely goes out of the disc is determined by $\sqrt{2m}l_B\leq R$, which gives $m\leq \frac{R^2}{2l_B^2}$. So that's roughly how many eigenstates there are in the disc.
A: why $r_{max}=\sqrt{2m}l_B$? You can square $\psi$ and take derivative w.r.t to z. And assume z is positive since radius r is positive. Then $|z|^2$ is just $z^2$. You solve this equation and then you can obtain $r_{max}$. This procedure is just finding the maximum value.
Other parts of your problem has already been answered very well.
