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.[1] 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,

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

Here, ${\displaystyle s={\frac {S}{D}}}$ where S is a particular spacing and D is the mean distance between neighboring intervals.[2]

• 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 ${\displaystyle 2\times 2}$ 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 ${\displaystyle 2\times 2}$ case and a good approximation in general) is given by

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

References

1. Random Matrices By Madan Lal Mehta. p. 13.
2. Principles of Quantum Computation and Information. p. 406.