= Laughlin wavefunction
{c}
{title2=$\Psi_m=\mathcal N\prod_{i<j}(z_i-z_j)^m e^{-\sum_i|z_i|^2/(4\ell_B^2)}$}
{wiki}
<Robert Laughlin>'s 1983 <variational method> uses pairwise zeros to suppress close <Electron> encounters within the <lowest Landau level>. For aligned <spin>, odd $m$ gives <fermionic exchange symmetry>.<Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations by Robert Laughlin>[pp. 1395–1396, Eqs. (5)–(7)]{ref}
Here $\mathcal N$ enforces <wavefunction normalization>, $\ell_B$ is the <magnetic length>, and <complex coordinates> $z_j=x_j-iy_j$ describe <Electrons> in a <magnetic field> $B\widehat{\mathbf z}$, $B>0$, with the <symmetric gauge>. Reversing the orientation conjugates the <polynomial> without changing <probability densities>. The <Gaussian function> confines the <wavefunction>.
The exchange check is especially simple for two <Electrons>. Writing $w=z_1-z_2$, exchange sends $w\mapsto-w$, so $w^m\mapsto(-1)^mw^m$. An aligned two-<Electron> <spin> state is symmetric, requiring the spatial <wavefunction> to change sign. Hence $m=1,3,5,\ldots$ works, whereas $m=2$ would give the wrong sign. At a close encounter, the <probability density> vanishes as $|w|^{2m}$: $m=3$ imposes a sixth-order zero, stronger than the quadratic zero at $m=1$. This is the local geometric mechanism for avoiding a large repulsive <Coulomb potential energy>.
A useful large-$N$ counting outline follows directly from the <polynomial>. Each <complex coordinate> has maximum degree $m(N-1)$. A <lowest Landau level> orbital proportional to $z^k e^{-|z|^2/(4\ell_B^2)}$ has its areal <probability density> maximum at $r^2=2k\ell_B^2$, by differentiating its logarithm. This suggests droplet <area> $A\simeq2\pi mN\ell_B^2$, neglecting the boundary correction. Dividing $N/A$ by the <degeneracy of a Landau level> per <area>, $1/(2\pi\ell_B^2)$, gives <filling factor> $\nu\simeq1/m$. This counting is a motivation for the fraction, not an independent proof that the <wavefunction> minimizes the interacting <Hamiltonian>.
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