In quantum mechanics, an antisymmetrizer
Contents
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
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
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
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
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
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
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,
The antisymmetrizer commutes with any observable
If it were otherwise, measurement of
Connection with Slater determinant
In the special case that the wave function to be antisymmetrized is a product of spin-orbitals
the Slater determinant is created by the antisymmetrizer operating on the product of spin-orbitals, as below:
The correspondence follows immediately from the Leibniz formula for determinants, which reads
where B is the matrix
To see the correspondence we notice that the fermion labels, permuted by the terms in the antisymmetrizer, label different columns (are second indices). The first indices are orbital indices, n1, ..., nN labeling the rows.
Example
By the definition of the antisymmetrizer
Consider the Slater determinant
By the Laplace expansion along the first row of D
so that
By comparing terms we see that
Intermolecular antisymmetrizer
One often meets a wave function of the product form
and
Here
Typically, one meets such partially antisymmetric wave functions in the theory of intermolecular forces, where
The total system can be antisymmetrized by the total antisymmetrizer
where τ is a left coset representative. Since
we can write
The operator
so that we see that it suffices to act with