mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

, the Wigner surmise is a statement about the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

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 Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

. The surmise was a result of Wigner's introduction of

random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...

in the field of

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...

. The surmise consists of two postulates: *In a simple sequence (

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

and parity are same), the

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

for a spacing is given by, ::

p_w(s) = \frac e^.

: Here,

s = \frac

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

M

, with elements that are independent standard gaussian random variables, with joint distribution proportional to ::

e^=e^=e^.

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) with distribution proportional to

e^

, is given by ::

p_w(s) = \frac e^.

References

Mathematical physics Nuclear physics {{math-physics-stub

See also

References