4
1 Systems of Identical Particles
Using this notation, the wave function takes the form
(ξ 1 , . . . , ξ N ) = =ξ N | . . . ξ 1 |
(1.4)
Integration with respect to the ξ-variables is defined according to
. . .
dξ 1 . . . dξ N =
σ 1
· · ·
σ N
. . .
dx 1 . . . dx N
(1.5)
To discuss the permutation symmetry, let us introduce permutation operators,
defined according to
ˆ
P(ξ 1 , . . . , ξ N ) = (ξ P(1) , . . . , ξ P(N ) )
(1.6)
where P denotes a permutation of the figures 1, . . . , N :
P : i → P(i), i = 1, 2, . . . , N
(1.7)
The set of permutations of N elements forms a group, referred to as the symmetric
group S N , and so does the corresponding set of permutation operators. (Here, the
group multiplication is the successive application, ˆ
P
= ˆ
P ˆ
P
.) The permutation
operators are unitary, that is, ˆ
P
−1
= ˆ
P
† (see Exercise 1.1). Each permutation can
be obtained as a product (consecutive application) of transpositions (exchanging
two figures). While this way of generating permutations is not unique, the number
of transpositions involved is either always even or always odd, depending on the
respective permutation. In that sense, a permutation P is said to be “even” or “odd,”
and a corresponding sign of P can be defined according to
(−1)
P
=
+1, even number of transpositions
−1, odd number of transpositions
(1.8)
The symmetrization postulate of quantum theory states that the wave functions
for a system of N uniform particles must be either totally symmetric (bosons) or
totally antisymmetric (fermions) with respect to any permutation of the particle variables:
ˆ
P(ξ 1 , . . . , ξ N ) =
(−1)
P
(ξ 1 , . . . , ξ N ), fermions
(ξ 1 , . . . , ξ N ), bosons
(1.9)
In the following, we shall deal exclusively with fermions, specifically electrons. Here,
an immediate consequence of the symmetrization postulate is the Pauli principle,
stating that the wave function vanishes whenever the coordinates and spins of two
(or more) fermions coincide:
(ξ 1 , . . . , ξ N ) = 0 for ξ i = ξ j (i = j)
(1.10)
1 Systems of Identical Particles
Using this notation, the wave function takes the form
(ξ 1 , . . . , ξ N ) = =ξ N | . . . ξ 1 |
(1.4)
Integration with respect to the ξ-variables is defined according to
. . .
dξ 1 . . . dξ N =
σ 1
· · ·
σ N
. . .
dx 1 . . . dx N
(1.5)
To discuss the permutation symmetry, let us introduce permutation operators,
defined according to
ˆ
P(ξ 1 , . . . , ξ N ) = (ξ P(1) , . . . , ξ P(N ) )
(1.6)
where P denotes a permutation of the figures 1, . . . , N :
P : i → P(i), i = 1, 2, . . . , N
(1.7)
The set of permutations of N elements forms a group, referred to as the symmetric
group S N , and so does the corresponding set of permutation operators. (Here, the
group multiplication is the successive application, ˆ
P
= ˆ
P ˆ
P
.) The permutation
operators are unitary, that is, ˆ
P
−1
= ˆ
P
† (see Exercise 1.1). Each permutation can
be obtained as a product (consecutive application) of transpositions (exchanging
two figures). While this way of generating permutations is not unique, the number
of transpositions involved is either always even or always odd, depending on the
respective permutation. In that sense, a permutation P is said to be “even” or “odd,”
and a corresponding sign of P can be defined according to
(−1)
P
=
+1, even number of transpositions
−1, odd number of transpositions
(1.8)
The symmetrization postulate of quantum theory states that the wave functions
for a system of N uniform particles must be either totally symmetric (bosons) or
totally antisymmetric (fermions) with respect to any permutation of the particle variables:
ˆ
P(ξ 1 , . . . , ξ N ) =
(−1)
P
(ξ 1 , . . . , ξ N ), fermions
(ξ 1 , . . . , ξ N ), bosons
(1.9)
In the following, we shall deal exclusively with fermions, specifically electrons. Here,
an immediate consequence of the symmetrization postulate is the Pauli principle,
stating that the wave function vanishes whenever the coordinates and spins of two
(or more) fermions coincide:
(ξ 1 , . . . , ξ N ) = 0 for ξ i = ξ j (i = j)
(1.10)