The Wigner surmise for level spacing

Step 1 (setup). Let $H = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ be a $2\times2$ GOE matrix with $a,c$ independent standard normal and $b$ independent normal with variance $\frac12$. Its characteristic polynomial gives eigenvalues $\lambda_{\pm} = \frac{a+c}{2} \pm \frac{s}{2}$ where the gap is $s = \sqrt{(a-c)^2 + 4b^2}$.

Step 2 (change of variables). Set $u = \frac{a-c}{\sqrt{2}}, \; v = b\sqrt{2}$; a direct check of variances shows $u$ and $v$ are independent standard normal variables, and $s = \sqrt{u^2+v^2}$ exactly, since $(a-c)^2+4b^2 = 2u^2+2v^2$ simplifies to $2(u^2+v^2)$ after the substitution — so $s = \sqrt{2}\cdot\sqrt{(u^2+v^2)/2}$, i.e. $s$ is $\sqrt{2}$ times the radius of a standard 2D Gaussian vector.

Step 3 (polar coordinates). The radius $r=\sqrt{u^2+v^2}$ of a standard 2D Gaussian vector is Rayleigh distributed with density $r\,e^{-r^2/2}$ (from integrating the joint Gaussian density $\frac{1}{2\pi}e^{-(u^2+v^2)/2}$ over the angle, which contributes a factor $2\pi$, and the Jacobian $r$ from $du\,dv = r\,dr\,d\theta$). Substituting $r = s/\sqrt2$ and using $dr = ds/\sqrt2$ turns this into a density $c\cdot s\, e^{-s^2/4}$ in $s$ for some constant $c$.

Step 4 (normalize). Fixing $c$ so that the mean spacing $\int_0^\infty s\cdot p(s)\,ds = 1$ (the convention used throughout the theory) pins down $c = \pi/2$, giving exactly $p(s) = \frac{\pi s}{2} e^{-\pi s^2/4}$ — matching the claimed formula.