20
2 Second Quantization
F = H 0 ⊕ H 1 ⊕ H
A
2 ⊕ · · ·
(2.1)
being the direct sum of the N -electron Hilbert spaces, N = 0, 1, . . . , where for
N ≥ 2 we may confine us to the subspace of antisymmetric states, H
A
N . For N = 0,
the Hilbert space H 0 is spanned by exactly one state, referred to as the vacuum state,
|∅.
Obviously, F is a linear vector space, with a scalar product defined within each of
the linear subspaces. The definition of the scalar product can easily be generalized,
q
1 . . . q
N |q 1 . . . q N = 0
if N = N
(2.2)
to cover products of states with differing electron numbers.
Having established the mathematical background, we are in the position to introduce “fermion operators,” more specifically, creation and destruction operators.
Let us first consider the creation operators, which can be defined by specifying their
action on the Fock-space basis states:
c
†
q |q 1 . . . q N = |q 1 . . . q N q
(2.3)
Acting on a product state of N electrons, c
†
q generates a product state of N + 1
electrons, “creating” an electron in the one-particle state q. Note that |q 1 . . . q N q = 0
unless q = q 1 , . . . , q N . Beginning with
c
†
q |∅ = |q
(2.4)
an N -electron product state can be generated according to
|q 1 . . . q N = c
†
q N
. . . c
†
q 1
|∅
(2.5)
by letting the creation operators act successively on the vacuum state.
Next, let us consider the effect of the hermitian conjugate operator, c q , that is,
c q = (c
†
q )
† , acting on an N -electron product state. For this purpose, we expand the
state of interest in terms of the Fock-space basis states:
c q |q 1 . . . q N =
q
1 2 ··· N
N
=0,1,...
|q
1 . . . q
N q
1 . . . q
N |c q |q 1 . . . q N
=
q
1 2 ··· N
N
=0,1,...
|q
1 . . . q
N q 1 . . . q N |c
†
q |q
1 . . . q
N
∗
(2.6)
In the second line, we have used the relation | ˆ
B| = =| ˆ
B
†
|
∗ to replace c q
by c
†
q . Since
c
†
q |q
1 . . . q
N = |q
1 . . . q
N q