(also known as an antisymmetrizing operator[1]) 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.
the wave function satisfies the Pauli exclusion principle.
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.
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.
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
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).
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.
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 antisymmetrizer carries a left and a right representation of the group: with the operator
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
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
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).
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 One often meets a wave function of the product form
The operators appearing in these two antisymmetrizers represent the elements of the subgroups SNA and SNB, respectively, of SNA+NB.
Typically, one meets such partially antisymmetric wave functions in the theory of intermolecular forces, where
When A and B interact, the Pauli principle requires the antisymmetry of the total wave function, also under intermolecular permutations.
However, in this way one does not take advantage of the partial antisymmetry that is already present.
It is more economic to use the fact that the product of the two subgroups is also a subgroup, and to consider the left cosets of this product group in SNA+NB: where τ is a left coset representative.