Elements of Modern Physics
134
with the total energy being
E =
1
i
N
a
i
E
=
∑
(5.10)
It is clear that any permutation of the particle indices in the solution in
Eq. (5.8) will again give an eigenstate of the Hamiltonian, with energy E. Since
all these states are degenerate, any linear combination of these states will also
be an eigenstate with energy E. In particular, the symmetric and antisymmetric
eigenstates are respectively
ψ + (1, 2, ..., N) =
1
2
1/ 2
perm
1
(1)
(2) ...
( )
( !)
ψ
ψ
ψ
∑
N
a
a
a
N
N
(5.11)
ψ – (1, 2, ..., N) =
1
1
1
2
2
2
1/ 2
(1)
(2)... ( )
(1) (2)... ( )
1 det.
...
...
( !)
(1)
(2)...
( )
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
N
N
N
a
a
a
a
a
a
a
a
a
N
N
N
N
(5.12)
where the summation is over all the permutations of the particle indices (the
normalization is different if some of the a i are the same). For the simple case of
two identical particles, the symmetric and antisymmetric states are
ψ ± (1, 2) =
1
2
1
2
1/ 2
1 [
(1)
(2)
(2)
(1)]
2
ψ
ψ
±ψ
ψ
a
a
a
a
(5.13)
In these equations, i.e. Eqs. (5.11) to (5.13), the solutions ψ + with the plus
sign are applicable to bosons while the solutions ψ – with the minus sign apply to
fermions. These solutions are of great importance as solutions for N identical
particles, and serve as approximate starting solutions even for systems whose
Hamiltonian cannot be written in the separable form in Eq. (5.7).
An extremely important point to note in Eqs. (5.12) and (5.13), is that the
fermion wave functions vanish if any two of the a i are equal. This means that
no two identical, noninteracting fermions can be in states described by the same
set of quantum numbers. This rule was first stated for electrons in an atom by
Pauli (1925), no two electrons in an atom can have the same set of quantum
numbers n, l, m l and m s , and is known as Pauli’s exclusion principle. It is central
to the understanding of the structure of atoms. We now begin the analysis of the
structure and the energy levels of an atom, subject to the constraints of Pauli’s
exclusion principle.
134
with the total energy being
E =
1
i
N
a
i
E
=
∑
(5.10)
It is clear that any permutation of the particle indices in the solution in
Eq. (5.8) will again give an eigenstate of the Hamiltonian, with energy E. Since
all these states are degenerate, any linear combination of these states will also
be an eigenstate with energy E. In particular, the symmetric and antisymmetric
eigenstates are respectively
ψ + (1, 2, ..., N) =
1
2
1/ 2
perm
1
(1)
(2) ...
( )
( !)
ψ
ψ
ψ
∑
N
a
a
a
N
N
(5.11)
ψ – (1, 2, ..., N) =
1
1
1
2
2
2
1/ 2
(1)
(2)... ( )
(1) (2)... ( )
1 det.
...
...
( !)
(1)
(2)...
( )
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
N
N
N
a
a
a
a
a
a
a
a
a
N
N
N
N
(5.12)
where the summation is over all the permutations of the particle indices (the
normalization is different if some of the a i are the same). For the simple case of
two identical particles, the symmetric and antisymmetric states are
ψ ± (1, 2) =
1
2
1
2
1/ 2
1 [
(1)
(2)
(2)
(1)]
2
ψ
ψ
±ψ
ψ
a
a
a
a
(5.13)
In these equations, i.e. Eqs. (5.11) to (5.13), the solutions ψ + with the plus
sign are applicable to bosons while the solutions ψ – with the minus sign apply to
fermions. These solutions are of great importance as solutions for N identical
particles, and serve as approximate starting solutions even for systems whose
Hamiltonian cannot be written in the separable form in Eq. (5.7).
An extremely important point to note in Eqs. (5.12) and (5.13), is that the
fermion wave functions vanish if any two of the a i are equal. This means that
no two identical, noninteracting fermions can be in states described by the same
set of quantum numbers. This rule was first stated for electrons in an atom by
Pauli (1925), no two electrons in an atom can have the same set of quantum
numbers n, l, m l and m s , and is known as Pauli’s exclusion principle. It is central
to the understanding of the structure of atoms. We now begin the analysis of the
structure and the energy levels of an atom, subject to the constraints of Pauli’s
exclusion principle.