Wigner surmise

In mathematical physics, the Wigner surmise is a statement about the probability distribution of the spaces between points in the spectra of nuclei of heavy atoms. It was proposed by Eugene Wigner in probability theory. The surmise was a result of Wigner's introduction of random matrices in the field of nuclear physics. The surmise consists of two postulates:

• In a simple sequence (spin and parity are same), the probability density function for a spacing is given by,

$p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.$

Here, $s={\frac {S}{D}}$ where S is a particular spacing and D is the mean distance between neighboring intervals.

• In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimposing simple sequences.

The above result is exact for $2\times 2$ real symmetric matrices, with elements that are independent and identically distributed standard gaussian random variables. In practice, it is a good approximation for the actual distribution for real symmetric matrices of any dimension. The corresponding result for complex hermitian matrices (which is also exact in the $2\times 2$ case and a good approximation in general) is given by

$p_{w}(s)={\frac {32s^{2}}{\pi ^{2}}}e^{-4s^{2}/\pi }.$

See also

• Wigner semicircle distribution