antisymmetric
Adjective
Meaning
(set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ∈ S, if both xRy and yRx then x=y.
(of a tensor) That changes sign when any two indices are interchanged (e.g., Tijk = -Tjik);
(of a bilinear form) For which B(w,v) = -B(v,w).
(linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof:
(of a matrix) Whose transpose equals its negative (i.e., M = −M);
Synonyms
Related terms
Examples
All 3 components of the dipole operator are antisymmetric under inversion with respect to the nucleus, : where is the dipole operator and is the inversion operator.
As a result, it can be regarded as an observable of the system, which means that, in principle, a measurement can be performed to find out if a state is symmetric or antisymmetric.
A more rigorous statement is that the total wave function for two identical fermions is antisymmetric with respect to exchange of the particles.
Approximated wavefunction of the deuteron The deuteron wavefunction must be antisymmetric if the isospin representation is used (since a proton and a neutron are not identical particles, the wavefunction need not be antisymmetric in general).
Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component : is necessarily antisymmetric.
Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric.