Wigner surmise

{{short description|Scientific hypothesis in mathematical physics}}

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, which have many degrees of freedom, or quantum systems with few degrees of freedom but chaotic classical dynamics. It was proposed by Eugene Wigner in probability theory.{{cite book |title=Random Matrices By Madan Lal Mehta |page=13 |isbn=9780080474113 |url=https://books.google.com/books?id=Kp3Nx03_gMwC&dq=wigner+surmise&pg=PA13|last1=Mehta |first1=Madan Lal |date=6 October 2004 |publisher=Elsevier }} 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,

:: $p\_w(s)\; =\; \backslash frac\{\backslash pi\; s\}\{2\}\; e^\{-\backslash pi\; s^2/4\}.$

: Here, $s\; =\; \backslash frac\{S\}\{D\}$ where S is a particular spacing and D is the mean distance between neighboring intervals.{{cite book |title=Principles of Quantum Computation and Information |page=406 |isbn=9789812563453 |url=https://books.google.com/books?id=9cRl0dFaeGMC&dq=wigner+surmise&pg=PA406|last1=Benenti |first1=Giuliano |last2=Casati |first2=Giulio |last3=Strini |first3=Giuliano |year=2004 |publisher=World Scientific }}

• 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 $2\backslash times\; 2$ real symmetric matrices $M$, with elements that are independent standard gaussian random variables, with joint distribution proportional to

::

\\\end{array}\right)^2}=e^{-\frac{1}{2}a^2-\frac{1}{2}c^2-b^2}.

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 $2\backslash times\; 2$ case and a good approximation in general) with distribution proportional to $e^\{-\backslash frac\{1\}\{2\}\{\backslash rm\; Tr\}(MM^\backslash dagger)\}$, is given by

:: $p\_w(s)\; =\; \backslash frac\{32\; s^2\}\{\backslash pi^2\}\; e^\{-4s^2/\backslash pi\}.$