In quantum mechanics, an antisymmetrizer (also known as antisymmetrizing operator) is a linear operator that makes a wave function of N identical fermions antisymmetric under the exchange of the coordinates of any pair of fermions. After application of the wave function satisfies the Pauli exclusion principle. Since is a projection operator, application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect, acting as the identity operator.
Mathematical definition
Consider a wave function depending on the space and spin coordinates of N fermions:
where the position vector ri of particle i is a vector in and σi takes on 2s+1 values, where s is the half-integral intrinsic spin of the fermion. For electrons s = 1/2 and σ can have two values ("spin-up": 1/2 and "spin-down": −1/2). It is assumed that the positions of the coordinates in the notation for Ψ have a well-defined meaning. For instance, the 2-fermion function Ψ(1,2) will in general be not the same as Ψ(2,1). This implies that in general and therefore we can define meaningfully a transposition operator that interchanges the coordinates of particle i and j. In general this operator will not be equal to the identity operator (although in special cases it may be).
A transposition has the
parity (also known as signature) −1. The Pauli principle postulates that a wave function of identical fermions must be an eigenfunction of a transposition operator with its parity as eigenvalue
Here we associated the transposition operator with the permutation of coordinates π that acts on the set of N coordinates. In this case π = (ij), where (ij) is the cycle notation for the transposition of the coordinates of particle i and j.
Transpositions may be composed (applied in sequence). This defines a product between the transpositions that is associative. It can be shown that an arbitrary permutation of N objects can be written as a product of transpositions and that the number of transposition in this decomposition is of fixed parity. That is, either a permutation is always decomposed in an even number of transpositions (the permutation is called even and has the parity +1), or a permutation is always decomposed in an odd number of transpositions and then it is an odd permutation with parity −1. Denoting the parity of an arbitrary permutation π by (−1)π, it follows that an antisymmetric wave function satisfies
where we associated the linear operator with the permutation π.
The set of all N! permutations with the associative product: "apply one permutation after the other", is a group, known as the permutation group or symmetric group, denoted by SN. We define the antisymmetrizer as
Properties of the antisymmetrizer
In the representation theory of finite groups the antisymmetrizer is a well-known object, because the set of parities forms a one-dimensional (and hence irreducible) representation of the permutation group known as the antisymmetric representation. The representation being one-dimensional, the set of parities form the character of the antisymmetric representation. The antisymmetrizer is in fact a character projection operator and is quasi-idempotent,
This has the consequence that for any N-particle wave function Ψ(1, ...,N) we have
Either Ψ does not have an antisymmetric component, and then the antisymmetrizer projects onto zero, or it has one and then the antisymmetrizer projects out this antisymmetric component Ψ'. The antisymmetrizer carries a left and a right representation of the group:
with the operator representing the coordinate permutation π. Now it holds, for any N-particle wave function Ψ(1, ...,N) with a non-vanishing antisymmetric component, that
showing that the non-vanishing component is indeed antisymmetric.
If a wave function is symmetric under any odd parity permutation it has no antisymmetric component. Indeed, assume that the permutation π, represented by the operator , has odd parity and that Ψ is symmetric, then
As an example of an application of this result, we assume that Ψ is a spin-orbital product. Assume further that a spin-orbital occurs twice (is "doubly occupied") in this product, once with coordinate k and once with coordinate q. Then the product is symmetric under the transposition (k, q) and hence vanishes. Notice that this result gives the original formulation of the Pauli principle: no two electrons can have the same set of quantum numbers (be in the same spin-orbital).
Permutations of identical particles are unitary, (the Hermitian adjoint is equal to the inverse of the operator), and since π and π−1 have the same parity, it follows that the antisymmetrizer is Hermitian,