= Two-electron Laughlin relative probability
{title2=$p_m(u)=\frac{u^{2m+1}e^{-u^2/4}}{2^{2m+1}m!}$}
Consider the original two-<Electron> example obtained by taking $N=2$ in the <Laughlin wavefunction>, with no additional <polynomial> for the <centre of mass>. Introduce $Z=(z_1+z_2)/2$, relative <complex coordinate> $w=z_1-z_2$, separation $r=|w|$, and dimensionless separation $u=r/\ell_B$. Since $|z_1|^2+|z_2|^2=2|Z|^2+|w|^2/2$, the <probability density> factorizes:
$$
|\Psi_m|^2\propto e^{-|Z|^2/\ell_B^2}\,r^{2m}e^{-r^2/(4\ell_B^2)}.
$$
Integrating out $Z$ leaves an isotropic relative <probability density>. Define $D_m(u)$ per unit dimensionless <area> and $p_m(u)$ per unit dimensionless separation:
$$
D_m(u)=\frac{u^{2m}e^{-u^2/4}}{\pi4^{m+1}m!},
\qquad p_m(u)=2\pi uD_m(u),
\qquad \int_0^\infty p_m(u)\,du=1.
$$
The physical areal <probability density> is $D_m(r/\ell_B)/\ell_B^2$. The probability of a separation between $r$ and $r+dr$ is $p_m(r/\ell_B)\,dr/\ell_B$. The extra $u$ comes from the <Jacobian determinant> in <polar coordinates>: a thin annulus has <area> $2\pi r\,dr$. Omitting that factor changes the question being answered.
\Image[/laughlin-two-electron-separation.png]
{title=Pair avoidance and the annular <Jacobian determinant> in a two-<Electron> <Laughlin wavefunction>}
{description=Original normalized model curves for chosen odd exponents $m=1$ and $m=3$. Left: relative <probability density> per unit dimensionless <area>. Right: <probability density> of dimensionless separation, including the annular factor $2\pi u$. Markers identify the respective maxima. These finite two-particle curves are not measured data or the pair distribution of an infinite fluid.}
Both <wavefunction normalization> and the peak shift can be checked exactly. The substitution $t=u^2/4$ gives the <Gamma function> integral
$$
\int_0^\infty u^{2m+1}e^{-u^2/4}\,du
=2^{2m+1}\int_0^\infty t^me^{-t}\,dt
=2^{2m+1}m!.
$$
For $m>0$, logarithmic differentiation gives
$$
\frac{d}{du}\log D_m=\frac{2m}{u}-\frac u2,
\qquad
\frac{d}{du}\log p_m=\frac{2m+1}{u}-\frac u2.
$$
Consequently the areal maximum lies at $u=2\sqrt m$, while the most probable separation lies farther out, at $u=\sqrt{4m+2}$. For $m=1$ these are $2$ and $\sqrt6\simeq2.449$; for $m=3$, $\sqrt{12}\simeq3.464$ and $\sqrt{14}\simeq3.742$. The same substitution gives $\langle u^2\rangle=4(m+1)$, so the <root mean square> separations are $\sqrt8$ and $4$ respectively. The distinction between a <probability density> maximum and a radial maximum therefore has a directly calculable geometric origin.
This two-<Electron> model explains exchange and pair avoidance. It does not establish a nonzero many-particle <spectral gap>, an extended <quantum Hall plateau>, or the topological properties of a fluid in the <thermodynamic limit>. Those require additional many-<Electron> reasoning and experimental tests.
Back to article page
