Antisymmetrizer
Encyclopedia
In quantum mechanics
, an antisymmetrizer
(also known as antisymmetrizing operator ) is a linear operator that makes a wave function of N identical
fermion
s antisymmetric under the exchange of the coordinates of any pair of fermions. After application of
the wave function satisfies the Pauli principle. Since 
is a projection operator,
application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect,
is then equal to the identity operator.
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 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. After this preamble we are ready to give the definition of the antisymmetrizer
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 therefore 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,
The antisymmetrizer commutes with any observable
(Hermitian operator corresponding to a physical—observable—quantity)
If it were otherwise, measurement of
could distinguish the particles, in contradiction with the assumption that only the coordinates of indistinguishable particles are affected by the antisymmetrizer.
the antisymmetrizer yields a constant times a Slater determinant
:
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.
Consider the unnormalized Slater determinant
By the Laplace expansion
along the first row of D
so that
By comparing terms we see that
where the total wave function is not antisymmetric, but the factors are antisymmetric,
and
Here
antisymmetrizes the first NA particles and 
antisymmetrizes the second set of NB particles.
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
is the electronic wave function of molecule A and 
is the wave function of molecule B. When A and B interact, the Pauli principle requires the antisymmetry of the total wave function, also under intermolecular permutations.
The total system can be antisymmetrized by the total antisymmetrizer
which consists of the (NA + NB)! terms in the group SNA+NB. 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 coset
s of this product group in SNA+NB:
where τ is a left coset representative. Since
we can write
The operator
represents the coset representative τ (an intermolecular coordinate permutation). Obviously the intermolecular antisymmetrizer 
has a factor NA! NB! less terms then the total antisymmetrizer.
Finally,
so that we see that it suffices to act with
if the wave functions of the subsystems are already antisymmetric.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, an antisymmetrizer
(also known as antisymmetrizing operator ) is a linear operator that makes a wave function of N identicalIdentical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s antisymmetric under the exchange of the coordinates of any pair of fermions. After application of
the wave function satisfies the Pauli principle. Since is a projection operator,application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect,
is then equal to 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 spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
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 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 permutationPermutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
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
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
.
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
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
, denoted by SN. After this preamble we are ready to give the definition of the antisymmetrizer
Properties of the antisymmetrizer
In the representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
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 characterCharacter theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of the antisymmetric representation. The antisymmetrizer is in fact a character projection operator and is therefore 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, thenAs 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
Unitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
, (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
(Hermitian operator corresponding to a physical—observable—quantity)If it were otherwise, measurement of
could distinguish the particles, in contradiction with the assumption that only the coordinates of indistinguishable particles are affected by the antisymmetrizer.Connection with Slater determinant
In the special case that the wave function to be antisymmetrized is a product of spin-orbitalsthe antisymmetrizer yields a constant times a Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
:
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 antisymmetrizerConsider the unnormalized Slater determinant
By the Laplace expansion
Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of...
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 where the total wave function is not antisymmetric, but the factors are antisymmetric,and
Here
antisymmetrizes the first NA particles and antisymmetrizes the second set of NB particles.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
is the electronic wave function of molecule A and is the wave function of molecule B. When A and B interact, the Pauli principle requires the antisymmetry of the total wave function, also under intermolecular permutations.The total system can be antisymmetrized by the total antisymmetrizer
which consists of the (NA + NB)! terms in the group SNA+NB. 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 cosetCoset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of this product group in SNA+NB:
where τ is a left coset representative. Since
we can write
The operator
represents the coset representative τ (an intermolecular coordinate permutation). Obviously the intermolecular antisymmetrizer has a factor NA! NB! less terms then the total antisymmetrizer.Finally,
so that we see that it suffices to act with
if the wave functions of the subsystems are already antisymmetric.