Antisymmetric function
In mathematics, an antisymmetric function or skew-symmetric function is a function of several variables, in which the exchange of two variables reverses the sign of the function. Important special cases antisymmetric functions are antikommutative links and alternating multilinear forms . In quantum mechanics , fermions are precisely those particles whose wave function is antisymmetric with respect to the exchange of particle positions.
The counterpart to the antisymmetric functions are symmetric functions .
definition
If and are two vector spaces (mostly over the real or complex numbers), then a multivariate function is called antisymmetric if for all permutations and all vectors
holds, where is the sign of the permutation.
Examples
Concrete examples
The subtraction
is antisymmetric, because swapping the two operands and reverses the sign of the result. For example, antisymmetric functions of three variables are
or
- .
More general examples
- the cross product of two vectors is antisymmetric
- the Lie bracket of two vectors is also antisymmetric
- an anti- commutative two - digit combination is an anti-symmetric function of the two operands
- the determinant of a matrix is an antisymmetric function of the column vectors of the matrix
- an alternating multilinear form is an antisymmetric function in the scalar body that is linear in every argument
further criteria
To prove the antisymmetry of a function, not all possible permutations of the symmetric group have to be checked. Since every permutation can be written as a sequential execution of transpositions of the form , a function is already antisymmetric if and only if the function value is reversed by interchanging any two variables and , that is
for with is. For further possible criteria for verifying the antisymmetry, see Symmetrical functions that must be used with a change in sign.
properties
The antisymmetric functions form a subspace in the vector space of all functions from to (with the component-wise addition and scalar multiplication ), that is
- a scalar multiple of an antisymmetric function is again an antisymmetric function and
- the sum of two antisymmetric functions is also antisymmetric again,
where the null function is trivially antisymmetric.
Antisymmetrization
By antisymmetrization, that is, by a weighted summation over all possible permutations of the shape
every non-antisymmetric function can be assigned an associated antisymmetric function . The antisymmetrization operator carries out a projection onto the subspace of the antisymmetric functions. If a product is a product of functions that only depend on a single variable (in quantum chemistry such a function is called a Hartree product), one can also write as a Slater determinant .
See also
literature
- Ilka Agricola , Thomas Friedrich : Vector analysis: differential forms in analysis, geometry and physics . Springer, 2010, ISBN 3-8348-1016-9 .
Web links
- Robert Milson, Thomas Foregger, Joe Corneli: Antisymmetric Mapping . In: PlanetMath . (English)