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G
Quantum Mechanical Perturbation Theory
Quantum mechanical perturbation theory is a widely used method in solid-
state physics. Without the details of derivation, we shall list a number of basic
formulas of time-independent (stationary) and time-dependent perturbation
theory below. For simplicity, we shall use the Dirac notation for wavefunctions
and matrix elements.
G.1 Time-Independent Perturbation Theory
Assume that the complete solution (eigenfunctions and eigenvalues) of the
Schrödinger equation
H 0
∣ψ (0) i
= E
(0) i
∣ψ (0) i
(G.1.1)
is known for a system described by a simple Hamiltonian H 0. If the system
is subject to a time-independent (stationary) perturbation described by the
Hamiltonian H 1 – which can be an external perturbation or the interaction
between the components of the system –, the eigenvalues and eigenfunctions
change. The method for determining the new ones depends on whether the
unperturbed energy level in question is degenerate or not.
G.1.1 Nondegenerate Perturbation Theory
We now introduce a fictitious coupling constant λ, whose value will be treated
as a parameter in the calculations and set equal to unity in the final result,
and write the full Hamiltonian H = H 0 + H 1 as
H = H 0 + λH 1. (G.1.2)
The parameter λ is purely a bookkeeping device to keep track of the relative
order of magnitude of the various terms, since the energy eigenvalues and
eigenfunctions will be sought in the form of an expansion in powers of λ:
580 G Quantum Mechanical Perturbation Theory
∣ψ i
∣ψ (0) i
∞ ∑
n=
λ
n
∣ψ (n) i
Ei = E
(0) i
∞ ∑
n=
λ
n E
(n) i
(G.1.3)
The series is convergent if the perturbation is weak, that is, in addition to the
formally introduced parameter λ, the interaction Hamiltonian itself contains
a small parameter, the physical coupling constant.
By substituting this expansion into the Schrödinger equation and collect-
ing the same powers of λ from both sides, we obtain
H 0
∣ψ (0) i
= E
(0) i
∣ψ (0) i
H 0
ψ
(1) i
+ H 1
ψ
(0) i
= E
(0) i
ψ
(1) i
+ E
(1) i
ψ
(0) i
H 0
ψ
(2) i
+ H 1
ψ
(1) i
= E
(0) i
ψ
(2) i
+ E
(1) i
ψ
(1) i
+ E
(2) i
ψ
(0) i
(G.1.4)
and similar equations for higher-order corrections. The corrections to the en-
ergy and wavefunction of any order are related to the lower-order ones by the
recursion formula
(H 0 − E
(0) i )
ψ
(n) i
+ (H 1 − E
(1) i )
ψ
(n−1) i
− E
(2) i
∣ψ (n−2) i
−... − E
(n) i
∣ψ (0) i
(G.1.5)
Multiplying the second equation in (G.1.4) (which comes from the terms
that are linear in λ) by
ψ
(0) i
∣ (^) from the left, the first-order correction to the
energy is
E
(1) i
ψ
(0) i
∣H
1
∣ψ (0) i
. (G.1.6)
To determine the correction to the wavefunction, the same equation is multi-
plied by
ψ
(0) j
(j = i):
E
(0) j
ψ
(0) j
∣ψ (1) i
ψ
(0) j
∣H
1
∣ψ (0) i
〉 = E
(0) i
ψ
(0) j
∣ψ (1) i
. (G.1.7)
Since the eigenfunctions of H 0 make up a complete set, the functions
ψ
(n) i
can be expanded in terms of them:
∣ψ (n) i
j
C
(n) ij
∣ψ (0) j
. (G.1.8)
The coefficients C
(n) ii are not determined by the previous equations: their val-
ues depend on the normalization of the perturbed wavefunction. Substituting
the previous formula into (G.1.7), we have
E
(0) j
C
(n) ij
ψ
(0) j
∣H
1
∣ψ (0) i
〉 = E
(0) i
C
(n) ij
, (G.1.9)
582 G Quantum Mechanical Perturbation Theory
which projects onto the subspace that is orthogonal to the state
∣ψ (0) i
. The
nth-order energy correction can then be written as
E
(n) i =^
ψ
(0) i
H 1
ψ
(n−1) i
, (G.1.16)
where the matrix element is to be taken with the wavefunction
ψ
(n) i
E
(0) i
− H 0
Pi
[
H 1 − E
(1) i
ψ
(n−1) i
−E
(2) i
∣ψ (n−2) i
−... − E
(n−1) i
∣ψ (1) i
]
(G.1.17)
which is in the subspace mentioned above.
Formally simpler expressions can be obtained when the Brillouin–Wigner
perturbation theory is used. The perturbed wavefunction in the Schrödinger
equation ( H 0 + H 1
∣ψ i
= Ei
∣ψ i
(G.1.18)
is then chosen in the form
∣ψ i
= C 0
∣ψ (0) i
∣Δψ i
, (G.1.19)
where
Δψi
is orthogonal to
ψ
(0) i
, and C 0 takes care of the appropriate
normalization. After some algebra, the eigenvalue equation reads
H 0 − Ei
∣Δψ i
+ H 1
∣ψ i
= C 0
Ei − E
(0) i
∣ψ (0) i
. (G.1.20)
By applying the projection operator Pi, and exploiting the relations
Pi
∣ψ (0) i
= 0 , Pi
∣ψ i
∣Δψ i
(G.1.21)
as well as the commutation of Pi and H 0 ,
Ei − H 0
∣Δψ i
= PiH 1
∣ψ i
(G.1.22)
is obtained. Its formal solution is
ψi
= C 0
ψ
(0) i
Pi
Ei − H 0
H 1
ψi
. (G.1.23)
Iteration then yields
∣ψ i
= C 0
∞ ∑
n=
Pi
Ei − H 0
H 1
)n ∣ ∣ψ (0) i
, (G.1.24)
and
ΔEi =
∞ ∑
n=
ψ
(0) i
∣H
1
Pi
Ei − H 0
H 1
)n ∣ ∣ψ (0) i
(G.1.25)
G.1 Time-Independent Perturbation Theory 583
for the energy correction. In this method the energy denominator contains the
perturbed energy Ei rather than the unperturbed one E
(0) i
. To first order in
the interaction,
Ei = E
(0) i
ψ
(0) i
∣H
1
∣ψ (0) i
, (G.1.26)
while to second order,
Ei = E
(0) i +^
ψ
(0) i
H 1
ψ
(0) i
j =i
ψ
(0) i
H 1
ψ
(0) j
ψ
(0) j
H 1
ψ
(0) i
Ei − E
(0) j
(G.1.27)
It is easy to show that by rearranging the energy denominator and expanding
it as
Ei − H 0
E
(0) i − H^0 + ΔEi
E
(0) i − H^0
∞ ∑
n=
−ΔEi
E
(0) i − H^0
)n
, (G.1.28)
the results of the Rayleigh–Schrödinger perturbation theory are recovered.
The formulas of time-dependent perturbation theory can also be used to
determine the ground-state energy and wavefunction of the perturbed sys-
tem, provided the interaction is assumed to be turned on adiabatically. The
appropriate formulas are given in Section G.2.
G.1.2 Degenerate Perturbation Theory
In the previous subsection we studied the shift of nondegenerate energy levels
due to the perturbation. For degenerate levels a slightly different method has
to be used because the formal application of the previous formulas would yield
vanishing energy denominators.
Assuming that the ith energy level of the unperturbed system is p-fold
degenerate – that is, the same energy E
(0) i belongs to each of the states
∣ψ (0) i 1
ψ
(0) i 2
ψ
(0) ip
–, any linear combination of these degenerate eigenstates
is also an eigenstate of H 0 with the same energy. We shall use such linear
combinations to determine the perturbed states. We write the wavefunctions
of the states of the perturbed system that arise from the degenerate states as
ψ
k
cik
ψ
(0) ik
n=i
cn
ψ
(0) n
, (G.1.29)
where the ci k are of order unity, whereas the other coefficients cn that specify
the mixing with the unperturbed eigenstates whose energy is different from
E
(0) i are small, proportional to the perturbation. By substituting this form
into the Schrödinger equation, and multiplying both sides by
ψ
(0) ij
from the
left,
G.2 Time-Dependent Perturbation Theory 585
cij (t) = c
(0) ij (t) +
∞ ∑
r=
λ
r c
(r) ij (t) , (G.2.4)
where, naturally, the zeroth-order term is a constant:
c
(0) ij (t) = δij. (G.2.5)
Substituting this series expansion into the Schrödinger equation, we find
i
∂t
c
(r) ij (t) =
k
e
i(E
(0) j −E
(0) k )t/ℏ
ψ
(0) j
∣H
1 (t)
∣ψ (0) k
c
(r−1) ik (t). (G.2.6)
The explicit formulas for the first two terms obtained through iteration are
c
(1) ij (t) = −
i
∫^ t
t 0
ψ
(0) j
∣H
1 (t 1 )
∣ψ (0) i
e
i(E
(0) j −E
(0) i )t^1 /ℏdt 1 ,^ (G.2.7)
and
c
(2) ij (t) =
i
) 2 ∫t
t 0
dt 1
∫^ t^1
t 0
dt 2
k
ψ
(0) j
∣H
1 (t 1 )
∣ψ (0) k
e
i(E
(0) j −E
(0) k )t^1 /ℏ
×
ψ
(0) k
∣H
1 (t 2 )
∣ψ (0) i
e
i(E (0) k −E (0) i )t 2 /ℏ (G.2.8)
In the interaction picture the time dependence of an arbitrary operator O
is given by
Oˆ(t) = eiH^0 t/ℏOe−iH^0 t/ℏ^. (G.2.9)
Using this form for the Hamiltonian, which may have an intrinsic time de-
pendence as well, the first two coefficients c
(n) ij can be written in terms of the
operators
ˆ H 1 (t) = e
iH 0 t/ℏ H 1 (t)e
−iH 0 t/ℏ (G.2.10)
as
c
(1) ij (t) =^ −^
i
∫^ t
t 0
ψ
(0) j
H 1 (t 1 )
ψ
(0) i
dt 1 (G.2.11)
and
c
(2) ij (t) =
i
) 2 ∫t
t 0
dt 1
t 1 ∫
t 0
dt 2
k
ψ
(0) j
∣ Hˆ
1 (t 1 )
∣ψ (0) k
×
ψ
(0) k
∣ Hˆ
1 (t 2 )
∣ψ (0) i
(G.2.12)
Since the intermediate states
ψ
(0) k
constitute a complete set, the previous
formula simplifies to
586 G Quantum Mechanical Perturbation Theory
c
(2) ij (t) =
i
) 2 ∫t
t 0
dt 1
∫^ t^1
t 0
dt 2
ψ
(0) j
H 1 (t 1 )
H 1 (t 2 )
ψ
(0) i
. (G.2.13)
The same result is obtained when the double integral on the t 1 , t 2 plane is
evaluated in reverse order:
c
(2) ij (t) =
i
2 ∫t
t 0
dt 2
∫^ t
t 2
dt 1
ψ
(0) j
H 1 (t 1 )
H 1 (t 2 )
ψ
(0) i
, (G.2.14)
or by swapping the notation of the two time variables:
c
(2) ij (t) =
i
) 2 ∫t
t 0
dt 1
∫^ t
t 1
dt 2
ψ
(0) j
∣ Hˆ
1 (t 2 )
Hˆ
1 (t 1 )
∣ψ (0) i
. (G.2.15)
Using these two formulas, the coefficient can also be written as
c
(2) ij (t) =
i
∫^ t
t 0
dt 1
∫^ t
t 0
dt 2
ψ
(0) j
T
H 1 (t 1 )
H 1 (t 2 )
ψ
(0) i
, (G.2.16)
where T is the time-ordering operator, which orders the operators in a product
in descending order of their time argument. Its action can be written in terms
of the Heaviside step function as
T
H 1 (t 1 )
H 1 (t 2 )
= θ(t 1 − t 2 )
H 1 (t 1 )
H 1 (t 2 ) + θ(t 2 − t 1 )
H 1 (t 2 )
H 1 (t 1 ).
(G.2.17)
Generalizing this to arbitrary orders, and setting λ = 1,
cij (t) = δji+
∞ ∑
n=
i
)n ∫t
t 0
dt 1
∫^ t^1
t 0
dt 2...
tn− 1 ∫
t 0
dtn
×
ψ
(0) j
∣ Hˆ
1 (t 1 )
Hˆ
1 (t 2 )^...^
Hˆ
1 (tn)
∣ψ (0) i
(G.2.18)
or, in time-ordered form,
cij (t) = δji+
∞ ∑
n=
n!
i
)n ∫t
t 0
dt 1
∫^ t
t 0
dt 2...
∫^ t
t 0
dtn
×
ψ
(0) j
∣T
Hˆ
1 (t 1 )
Hˆ
1 (t 2 )^...^
Hˆ
1 (tn)
∣ψ (0) i
(G.2.19)
The time evolution of the wavefunction between times t 0 and t is therefore
governed by the operator S(t, t 0 ):
∣ ∣ ψ(t)
= S(t, t 0 )
ψ(t 0 )
, (G.2.20)
588 G Quantum Mechanical Perturbation Theory
in the interaction picture. Using the ground state
of energy E 0 of the
unperturbed system, the energy correction due to the perturbation is
ΔE =
∣H
1 S(0,^ −∞)
0
S(0, −∞)
〉 , (G.2.28)
and the wavefunction is
S(0, −∞)
∣S(0, −∞)
0
〉 . (G.2.29)
As J. Goldstone (1957) pointed out, the same result may be formu-
lated in a slightly different way. Considering a many-particle system with a
nondegenerate ground state, the contribution of each term in the perturba-
tion expansion can be represented by time-ordered diagrams that show the
intermediate states through which the system gets back to the ground state.
This representation contains terms in which some of the particles participat-
ing in the intermediate processes are in no way connected to the incoming
and outgoing particles. It can be demonstrated that the contributions of the
disconnected parts are exactly canceled by the denominator in (G.2.28) and
(G.2.29), so
ΔE =
∞ ∑
n=
∣H 1
E 0 − H 0
H 1
)n ∣ ∣ ∣Ψ 0
con
∞ ∑
n=
E 0 − H 0
H 1
)n ∣ ∣ Ψ 0
con
(G.2.30)
where the label “con” indicates that only the contribution of connected di-
agrams need to be taken into account. It should be noted that instead of
Goldstone’s time-ordered diagrams, the perturbation series for the ground-
state energy can also be represented in terms of Feynman diagrams, which
are more commonly used in the many-body problem. Only connected dia-
grams need to be considered in that representation, too.
Reference
- C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, John
Wiley & Sons, New York (1977).
H
Second Quantization
The quantum mechanical wavefunction is most often considered as the func-
tion of the space and time variables when the solutions of the Schrödinger
equation are sought. In principle, this approach is applicable even when the
system is made up of a large number of interacting particles. However, it is
then much more convenient to use the occupation-number representation for
the wavefunction. We shall introduce the creation and annihilation operators,
and express the Hamiltonian in terms of them, too.
H.1 Occupation-Number Representation
It was mentioned in Chapter 12 on the quantum mechanical treatment of
lattice vibrations that the eigenstates of the harmonic oscillator can be char-
acterized by the quantum number n that can take nonnegative integer values.
Using the linear combinations of the position variable x and its conjugate
momentum, it is possible to construct operators a
† and a that increase and
decrease this quantum number. We may say that when these ladder operators
are applied to an eigenstate, they create an additional quantum or annihi-
late an existing one. Consequently, these operators are called the creation and
annihilation operators of the elementary quantum or excitation. States can
be characterized by the number of quanta they contain – that is, by the oc-
cupation number. Using Dirac’s notation, the state ψn of quantum number
n – which can be constructed from the ground state of the oscillator by the
n-fold application of the creation operator a † , and thus contains n quanta –
will henceforth be denoted by |n〉. The requirement that such states should
also be normalized to unity leads to
a|n〉 =
n |n − 1 〉 , a
† |n〉 =
n + 1 |n + 1〉. (H.1.1)
The operators a and a
† of the quantum mechanical oscillator satisfy the
bosonic commutation relation.
H.1 Occupation-Number Representation 591
We shall now define the creation and annihilation operators that act in
the Fock space and increase and decrease the occupation number by one:
a
† k |n 1 , n 2 ,... , nk,... 〉 = fc|n 1 , n 2 ,... , nk + 1,... 〉 ,
a k |n 1 , n 2 ,... , nk,... 〉 = fa|n 1 , n 2 ,... , nk − 1 ,... 〉.
(H.1.5)
If the normalization factors for bosons are chosen the same way as for harmonic
oscillators, that is,
a
† k |n 1 , n 2 ,... , nk,... 〉 =
nk + 1|n 1 , n 2 ,... , nk + 1,... 〉 ,
a k |n 1 , n 2 ,... , nk,... 〉 =
nk|n 1 , n 2 ,... , nk − 1 ,... 〉 ,
(H.1.6)
then nˆk = a
† k a k is the number operator that gives the occupation number of
the state of index k, since
a
† k a k |n 1 , n 2 ,... , nk,... 〉 = nk|n 1 , n 2 ,... , nk,... 〉 , (H.1.7)
and the commutation relations are the usual ones for bosons:
a k a
† k′^ − a
† k′^ a k = δkk′. (H.1.8)
Any state |n 1 , n 2 ,... , nk,... 〉 can be constructed from the vacuum by
means of creation operators:
Φn 1 ,n 2 ,...,nk ,... =
n 1 !n 2!... nk!...
a
† 1
)n 1
a
† 2
)n 2
...
a
† k
)n k
... | 0 〉. (H.1.9)
As has been mentioned, these states make up a complete set, and the wave-
functions of interacting many-particle system can be expressed as linear com-
binations of them.
Fermions
A similar approach can be adopted for fermions, however, the many-particle
wavefunction has to be chosen as
Φn 1 ,n 2 ,...,nk ,... =
N!
P
P φp 1 (ξ 1 )φp 2 (ξ 2 )... φpN (ξN) (H.1.10)
to meet the requirement of complete antisymmetrization. This is equivalent
to building a Slater determinant from the one-particle wavefunctions:
Φn 1 ,n 2 ,...,nk ,... =
N!
φp 1 (ξ 1 ) φp 2 (ξ 1 )... φpN (ξ 1 )
φp 1 (ξ 2 ) φp 2 (ξ 2 )... φpN (ξ 2 )
. . .
φp 1 (ξN) φp 2 (ξN)... φp N (ξN)
. (H.1.11)
592 H Second Quantization
Since each one-particle state can occur at most once, when the states are
ordered in some arbitrary way, the product in (H.1.10) with indices
p 1 < p 2 < · · · < pN (H.1.12)
is chosen with positive sign, and the signs for other configurations follow from
the parity of the permutation.
The occupation-number representation can be used for fermions as well.
The wavefunction is then written in Fock space as
Φn 1 ,n 2 ,...,n k ,...^ ≡ |n 1 , n 2 ,... , nk,... 〉 , (H.1.13)
where ni can be either 0 or 1. The creation and annihilation operators must
be introduced in such a way that the equations
a
† k |n 1 , n 2 ,... , nk,... 〉 = 0 , if nk = 1 ,
a k |n 1 , n 2 ,... , nk,... 〉 = 0 , if nk = 0
(H.1.14)
be satisfied. After ordering the one-particle states, the normalization of the
states obtained by the application of the creation and annihilation operators
are chosen as
a
† k |n 1 , n 2 ,... , nk,... 〉 =
1 − nk(−1)
Sk |n 1 , n 2 ,... , nk + 1,... 〉 ,
a k |n 1 , n 2 ,... , nk,... 〉 =
nk(−1)
Sk |n 1 , n 2 ,... , nk − 1 ,... 〉 ,
(H.1.15)
where
Sk =
i<k
ni. (H.1.16)
With this choice nˆk = a
† k a k is the number operator for fermions as well,
since when it acts on the state |n 1 , n 2 ,... , nk,... 〉
a
† k a k |n 1 , n 2 ,... , nk,... 〉 = nk|n 1 , n 2 ,... , nk,... 〉. (H.1.17)
In reverse order, however,
a k a
† k |n 1 , n 2 ,... , nk,... 〉 =
(1 + nk)(1 − nk)|n 1 , n 2 ,... , nk,... 〉. (H.1.18)
Since nk can only take the values 0 and 1, the eigenvalue of a k a
† k is 1 − nk,
and thus the operator identity
a k a
† k
† k a k
= 1 (H.1.19)
holds. By taking states of different quantum numbers, if the state of quantum
number k precedes the state with quantum number k
′ in the order, we have
a k a
† k′^ |n 1 , n 2 ,... , nk,... , nk′^ ,... 〉 (H.1.20)
Sk (−1)
Sk′
nk
1 − nk′^ |n 1 , n 2 ,... , nk − 1 ,... , nk′^ + 1,... 〉,
594 H Second Quantization
∗ n 1 ,n 2 ,...,nk ,...
N ∑
i=
f (ξi)Φn 1 ,n 2 ,...,nk ,...^ dξ 1 dξ 2... dξN
= N
n 1 !n 2!... nk!...
N!
l
φ
∗ l (ξ)f (ξ)φ l (ξ) dξ (H.2.2)
×
P ′
φ
∗ p 1 (ξ 2 )^... φ
∗ pN (ξN )φp 1 (ξ 2 )^... φpN (ξN ) dξ 2...^ dξN.
To calculate the factor that remains after the separation of the matrix element
of the state l, only those states need to be considered in the permutation P
′
that contain the state of label l only nl − 1 times. Owing to the orthonormality
of the one-particle states, the value of the previous formula is
l
nl
φ
∗ l (ξ)f (ξ)φ l (ξ) dξ. (H.2.3)
In the off-diagonal terms nonzero matrix elements are obtained between
those states Φn 1 ,n 2 ,...,nk ,...,nl,... and Φn 1 ,n 2 ,...,nk +1,...,nl− 1 ,... for which the oc-
cupation numbers of two one-particle states differ by one unit each. Then
∗ n 1 ,n 2 ,...,nk +1,...,nl− 1 ,...
i
f (ξi)Φn 1 ,n 2 ,...,nk ,...,nl,...^ dξ 1 dξ 2... dξN.
(H.2.4)
Because of the normalization factors of the two wavefunctions the matrix
element is proportional to
I =
n 1 !n 2!... (nk + 1)!... (nl − 1)!
N!
n 1 !n 2!... nk!... nl!
N!
(H.2.5)
Since each particle contributes by the same amount, the matrix element is
∗ n 1 ,n 2 ,...,nk +1,...,nl− 1 ,...
i
f (ξi)Φn 1 ,n 2 ,...,nk ,...,nl,... dξ 1 dξ 2... dξN
= N I
kl
φ
∗ k (ξ)f (ξ)φ l (ξ) dξ (H.2.6)
×
P ′
φ
∗ p 1 (ξ^2 )^... φ
∗ pN (ξN^ )φp 1 (ξ^2 )^... φpN (ξN^ ) dξ^2...^ dξN^.
After the separation of the integral for the selected particle, the remaining
terms correspond to a state that contains N − 1 particles, with occupation
numbers n 1 , n 2 ,... , nk,... , nl − 1 ,.... Since there are
(N − 1)!
n 1 !n 2!... nk!... (nl − 1)!...
(H.2.7)
H.2 Second-Quantized Form of Operators 595
such states, the separation of the ξ-integral leaves behind a factor
nk + 1
nl,
so the matrix element is
nk + 1
nl
φ
∗ k (ξ)f (ξ)φ l (ξ) dξ. (H.2.8)
The same expressions are obtained for the diagonal and off-diagonal matrix
elements if the states are specified in occupation-number representation, the
operator F
(1) is chosen as
F
(1)
kl
a
† k fkla l
, (H.2.9)
where
fkl =
φ
∗ k (ξ)f (ξ)φ l (ξ) dξ , (H.2.10)
and the previously obtained relations for the action of the creation and annihi-
lation operators are used in the calculation of the matrix element. Therefore
the operator given in (H.2.9), which acts in the Fock space, is the second-
quantized form of one-particle operators for bosons. Note that while the sum
is over N particles in the first-quantized formula (H.2.1) of the one-particle
operator, it is over the quantum numbers of the one-particle states in the
second-quantized formula.
The intermediate steps are slightly different for fermions, since a Slater
determinant wavefunction is specified in terms of the occupation numbers,
and the normalization factors are also different – nevertheless the final result
is the same: the one-particle operators for fermions can again be represented
as (H.2.9) in terms of creation and annihilation operators.
H.2.2 Second-Quantized Form of Two-Particle Operators
This approach can be extended to the two-body interaction term in the Hamil-
tonian and similar operators that are the sums of terms containing the coor-
dinates of two particles:
F
(2)
ij
f (ξi, ξj ). (H.2.11)
Since the variables of two particles appear in each term, such an operator has
a nonvanishing matrix element only between states for which the occupation
numbers of at most four one-particle states change: two decrease and two
others increase by one. The matrix element to be evaluated is thus
∗ n 1 ,...,nk +1,...,nl+1,...,nm− 1 ,...,nn− 1 ,...
ij
f (ξi, ξj )
Φn 1 ,...,nk ,...,nl,...,nm,...,nn,...^ dξ 1 dξ 2... dξN.
(H.2.12)
H.2 Second-Quantized Form of Operators 597
[
ψ(ξ),
ψ(ξ
′ )
]
∓
[
ψ
† (ξ),
ψ
† (ξ
′ )
]
∓
= 0. (H.2.18)
When the spin variable is separated,
[
ψα(r),
ψ
† β (r
′ )
]
∓
= δαβ δ(r − r
′ ) , (H.2.19)
and [ ˆ ψα(r),
ψ β (r
′ )
]
∓
[
ψ
† α(r),^
ψ
† β (r
′ )
]
∓
= 0. (H.2.20)
The creation and annihilation operators a
† k and a k change the occupation
of a one-particle state of a given quantum number. Below we shall see that
field operators can be interpreted as operators that create and annihilate a
particle at ξ. In other words, the state
ψ
† σ (r)|^0 〉^ contains a particle of spin
quantum number σ at point r of the real space. Similarly,
ψ
† σ (r)
ψσ (r) is the
density operator of spin-σ particles at r.
The one- and two-particle operators can then be rewritten as
F
(1)
dξ ψˆ
† (ξ)f
(1) (ξ) ψˆ(ξ) , (H.2.21)
and
F
(2)
dξ 1
dξ 2 ψˆ
† (ξ 1 ) ψˆ
† (ξ 2 )f
(2) (ξ 1 , ξ 2 ) ψˆ(ξ 2 ) ψˆ(ξ 1 ). (H.2.22)
In this representation the one-particle operator has exactly the same form as
the expectation value of the one-particle operator in first quantization, except
that the wavefunction and it complex conjugate are replaced by the field
operator and its Hermitian adjoint – hence the name second quantization.
H.2.4 Second-Quantized Form of the Electronic Hamiltonian
Among the terms of the Hamiltonian, the kinetic energy and the potential are
one-particle operators, and the pair interaction is a two-particle operator. For
electrons, when the potential and the pair interaction are spin independent,
H =
i
H
(1) i +^
1 2
ij
U
(2) (ri, rj ) , (H.2.23)
where the one-particle part contains
H
(1) i
2
2 me
2 i
By taking a complete set of one-particle states, and denoting, as customary
for electrons, the creation and annihilation operators of a particle in state
φk(ξ) by c
† k and c k instead of a
† k and a k , we have
598 H Second Quantization
H =
kl
Hklc
† k c l
1 2
klmn
U
(2) klmn c
† k c
† l c m c n
, (H.2.25)
where
Hkl =
φ
∗ k(ξ)
2
2 me
2
φl (ξ) dξ , (H.2.26)
and
U
(2) klmn
dξ dξ
′ φ
∗ k(ξ)φ
∗ l (ξ
′ )U
(2) (r, r
′ )φm(ξ
′ )φn(ξ). (H.2.27)
In general, the states are chosen in such a way that the one-particle part be
diagonal. This is the case when the Bloch functions determined in the presence
of a periodic potential are used as a complete basis set. However, this is not
the only option. In the Hubbard model the Wannier states are used, and so the
one-particle term in the Hamiltonian that describes the hopping of electrons
between lattice points is not diagonal.
Using the field operators instead of the creation and annihilation operators,
H =
ψ
† (ξ)
2
2 me
2
ψ(ξ) dξ
dξ dξ
ψ
† (ξ)
ψ
† (ξ
′ )U
(2) (r, r
′ )
ψ(ξ
′ )
ψ(ξ).
(H.2.28)
Writing out the spin variable explicitly, the spin independence of the potential
and of the interaction implies
H =
σ
ψ
† σ (r)
2
2 me
2
ψσ (r) dr
σσ′
dr dr
ψ
† σ (r)
ψ
† σ′^ (r
′ )U
(2) (r, r
′ )
ψ σ′^ (r
′ )
ψσ (r).
(H.2.29)
The description is highly simplified by choosing the plane waves as the
complete set. The one-particle states are then characterized by the wave vector
k and the spin quantum number σ. The usual formula
Hkin = −
i
2
2 me
2
∂r 2 i
(H.2.30)
for the kinetic energy can be rewritten in second-quantized form as
Hkin =
kk′σσ′
c
† kσ Hσσ′ (k, k
′ )c k′σ′^
, (H.2.31)
where
