Juan C Paniagua
580 California St., Suite 400
San Francisco, CA, 94104
Second Quantization Formalism ⇤
2017
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Abstract
Quantum chemists we are seldom concern with phenomena involving variations in the number of particles, so they do not need to resort to quantum field theories to lay the foundations of their work (one exception are certain spectroscopic phenomena with require a quantum description of electromagnetic radiation). The standard quantum mechanic theory for material particles, in which the number of particles of every type remains ⇤This document was writen en 2014 and revised in 2017 for the subject Mathematical Foundations of Quantum Mechanics of the Máster Interuniversitario en Química Teórica y Modelización Computacional. It is published under the Attribution 4.0 Creative
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Second Quantization Formalism⇤
Juan Carlos Paniagua
E-mail: [email protected]
Departament de Ciència de Materials i Química Física &
Institut de Química Teòrica i Computacional (IQTC-UB)
Universitat de Barcelona
December, 2014 - Revised on April 28th , 2018
Contents
1 Introduction 2
2 The Fock space 2
3 Electron creation and annihilation operators 2
3.1 Number operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Anticommutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 The non-relativistic many-electron hamiltonian in second quantization 6
4.1 Restricted spin-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Change of spin-orbital basis set 8
5.1 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Particles and holes 9
7 Non-fixed-particle systems 9
⇤ This document was written for the subject Mathematical Foundations of Quantum Mechanics of the Máster Interuniversi-
tario en Química Teórica y Modelización Computacional. It is published under the Attribution 4.0 Creative Commons Inter-
national License (CC BY 4.0 ). You can download it from the Dipòsit Digital of the Universitat de Barcelona
(http://hdl.handle.net/2445/115576). You are free to copy and redistribute the material, remix, transform, and build upon it
for any purpose, provided that you give appropriate credit, supply a link to the license, and indicate if changes were made. See
https://creativecommons.org/licenses/by/4.0/ for further details.
2 Juan Carlos Paniagua
1 Introduction
Material particles can be created or destroyed in processes such as the conversion of a gamma photon into an
electron-positron pair. To deal such phenomena we need to resort to quantum electrodynamics, that treats the
interplay between electrons, positrons and photons. In this theory operators that create or destroy particles
play a central role. In ordinary chemical processes the number of particles is kept constant, so no creation
and annihilation operators are needed to describe them.1 The standard quantum mechanic theory for material
particles, in which the number of particles of every type does not change over time, is then a suitable theoretical
framework. However, even in this context it is often convenient to use some tools of quantum field theories —
specifically, the creation and annihilation operators— to state certain mathematical developments, particularly
in the study of infinite systems. This way of formulating the theory is known as second quantization formalism.2
2 The Fock space
Creation and annihilation operators are applications that, when applied to a state of an n-particle system,
produce a state of an (n + 1)- and an (n 1)-particle system, respectively. Therefore they act in a broader
Hilbert space that those considered so far, which is known as the Fock space (F). If all of the variable-number
particles are of the same type the Fock space is the direct sum of every fixed-particle-number space.3
In the particular case of an electron system (or any system made of identical fermions) the Fock space is:
a a
F = H0 H1 H1⌦ 2
· · · H1⌦ n
···
where H0 and H1 are, respectively, the zero-electron and one-electron Hilbert spaces. H0 is a one-dimensional
space containing a normalized vector 0 = |i that represents a state with no electrons (the vacuum state), which
is different from the zero vector (0). Let us choose a normalized discrete basis set { 1 , · · · i , · · · } in H1 . The
set {n I } of all n-electron Slater determinants
n
↵
I ⌘ ( 1 I · · · nI )
a
is then a normalized basis of H1⌦ n , and the collection of all these basis for every value of n
0 1 2
, I , I · · · {n I} · · ·
is a normalized basis of F.
When referring to state vectors as elements of the Fock space an occupation-number representation is often
used, in which each basis vector n I is identified by a sequence of occupation numbers ni that take the value 1
for the spin-orbitals present in n I and 0 for all the other:
X
n
I = |n1 , · · · ni , · · · i with n= ni
i
↵
For instance, if 1 · · · n are the first n spin-orbitals of the one-electron basis set, then n 0 = ( 1 · · · n ) =
| 1, · · · 1, 0, 0, · · · i. For the vacuum state all the occupation numbers are zero: |i = |0, · · · 0, · · · i. In the case of
| {z }
n
boson systems the occupation numbers can take any natural value (including zero).
3 Electron creation and annihilation operators
The annihilation operator abi of an electron in the spin-orbital i is conveniently defined in the occupation-
number representation as
abi |n1 , · · · ni , · · · i = ( 1)⌫i ni |n1 , · · · 1 ni , · · · i
1 One exception are certain spectroscopic phenomena that require a quantum description of electromagnetic radiation, in which
photons can be created (radiation emission) and destroyed (radiation absorption).
2 While in quantum electrodynamics the energy associated to the classical electromagnetic fields becomes a quantized observable,
in the present formalism a certain type of quantization will emerge from the quantum wave functions, hence the term “second
quantization”.
3 One could wonder why to use direct products to build de Hilbert space of a many-particle system from the one-particle spaces
and direct sums to express the Fock space in terms of fixed-particle spaces. In the first case we have a complex system that can be
divided into different subsystems, and these are different from their union. On the other hand, the Fock space is the Hilbert space
of a single system in which the number of particles is not a fixed parameter, as in standard quantum mechanics, but an observable
that can take different values, may evolve in time, and may even not be well defined. In the Hilbert space containing the states of
this system there are subspaces that correspond to different eigenvalues of the “number of particles operator”, in the same way that
there are subspaces corresponding to different values of any other observable, and the direct sum of all of these subspaces gives the
whole Hilbert space.
Second Quantization Formalism (2014-2018) 3
Pi 1
where ⌫i = j=1 nj and ⌫1 = 0. The reason for the term “annihilation” will become clear by applying this
definition to some particular cases:
ab1 |1, n2 , · · · ni , · · · i = |0, n2 , · · · ni , · · · i
ab2 |0, 1, · · · ni , · · · i = |0, 0, · · · ni , · · · i
ab2 |1, 1, · · · ni , · · · i = |1, 0, · · · ni , · · · i
That is, if the spin-orbital i is occupied in the many-electron vector |n1 , · · · ni , · · · i then abi annihilates an
electron in that spin-orbital. So, from an n-electron vector we obtain an (n-1)-electron vector. Besides, it
changes the sign of the vector if i was in an even position among the occupied states. If i is empty in
|n1 , · · · ni , · · · i the result of applying abi to this vector is zero. Thus
ab1 |0, n2 , · · · ni , · · · i = 0
When we use the usual occupied-spin-orbitals-only notation for the Slater determinants then the effect of
the annihilation operator abi over a determinant containing i takes the form:
E E
abi ( j · · · i · · · k ) = ( 1)⌫i j ··· i··· k
where ⌫i is again the position number of i minus 1, and i means that i is absent in the determinant. ⌫i
is also the number of transpositions needed to bring i to the first position of the determinant.4 Since every
transposition introduces a change of sign, we can obtain the effect of abi by first bringing i to the first position
and then dropping it from the determinant.
For a determinant that does not contain i it is clear that
E
abi j ··· i ··· k =0
The creation operator abi † of an electron in the spin-orbital i is defined by
abi † |n1 , · · · ni , · · · i = ( 1)⌫i (1 ni ) |n1 , · · · 1 ni , · · · i
Some examples reveal that this operator creates an electron in the spin-orbital i if this was empty, and
introduces a change of sign if the creation takes place in an even position among the occupied spin-orbitals:
ab1 † |0, n2 , · · · ni , · · · i = |1, n2 , · · · ni , · · · i
†
ab2 |0, 0, · · · ni , · · · i = |0, 1, · · · ni , · · · i
†
ab2 |1, 0, · · · ni , · · · i = |1, 1, · · · ni , · · · i
Therefore, it produces an (n+1)-electron vector from an n-electron one. If i is occupied in |n1 , · · · ni , · · · i the
result of applying abi † to this vector is zero; for instance:
ab1 † |1, n2 , · · · ni , · · · i = 0
Any many-electron basis vector can be obtained from the vacuum state by successive application of creation
operators: ⇣ ⌘ n1 ⇣ ⌘ ni
|n1 , · · · ni , · · · i = ab1 † · · · abi † · · · |0, · · · 0, · · · i
In terms of the occupied-spin-orbitals-only notation the effect of the creation operator abi † over a determinant
not containing i is: E E
abi † j · · · i · · · k = ( 1) ⌫i
( j · · · i · · · k )
where ⌫i is the position in which i has been made to appear minus 1. ⌫i is also the number of transpositions
needed to bring i to the first position of the determinant:
E E E
abi † ( j · · · i 1 i+1 · · · k ) = ( 1)⌫i ( j · · · i 1 i i+1 · · · k ) = ( i j · · · i 1 i+1 · · · k )
so that we can also say that abi † creates an electron in the spin-orbital i placed in the first position of the
determinant: E E
abi † j ··· i··· k = ( i j · · · k)
4 When the Slater determinants are represented by sequences of occupation numbers the spin-orbital ordering must be the same
for all of them, but in the occupied-spin-orbitals-only notation this ordering can be altered.
4 Juan Carlos Paniagua
Then, if we want to advance it ⌫i positions we have to introduce ⌫i changes of sign; that is, we have to multiply
the determinant by ( 1)⌫i .
It is evident that E
abi † ( j · · · i · · · k ) =0
Let us now show that abi † is the adjoint of abi . From now on we will assume that the one-electron basis
{ 1 , · · · i , · · · } is orthonormal, although some of the results that will be obtained are independent of this
assumption.
We want to prove the equality
D E
hn01 , · · · n0i , · · · |abi | n1 , · · · ni , · · · i = abi † (n01 , · · · n0i , · · · ) n1 , · · · ni , · · ·
for any two sequences of occupation numbers {n01 , · · · n0i , · · · } and {n1 , · · · ni , · · · }. By using the above definition
of abi † the right hand side member becomes
0 0
( 1)⌫i (1 n0i ) h n01 , · · · 1 n0i , · · · | n1 , · · · ni , · · · i = ( 1)⌫i (1 n0i ) n0 ,n1
1
··· 1 n0i ,ni ···
Likewise, the left hand side member is
hn01 , · · · n0i , · · · |( 1)⌫i ni | n1 , · · · 1 ni , · · · i = ( 1)⌫i ni n0 ,n1
1
··· n0i ,1 ni ···
This two expressions vanish unless n01 = n1 , · · · n0i = 1 ni · · · , in which case they coincide.
Exercise ↵
Let = ( 1 · · · i · · · n ) be the Hartree-Fock Slater determinant of an n-electron system and let ki
be the determinant that results upon changing in the occupied spin-orbital i by an empty one k . The
spin-orbitals are assumed orthonormal.
• Write k
i in terms of
by applying on this the proper creation and annihilation operators.
⌦ ↵
• Use the resulting expression to show that k
i = 0.
3.1 Number operators
The product
abi † abi ⌘ nbi
is known as occupation number operator of the spin-orbital i for reasons that will now become evident:
abi † abi |n1 , · · · ni , · · · i = abi † ( 1)⌫i ni |n1 , · · · 1 ni , · · · i = ( 1)⌫i ni ( 1)⌫i (1 (1 ni )) |n1 , · · · ni , · · · i
Since ni can only take the values 1 and 0, n2i = ni and
nbi |n1 , · · · ni , · · · i = ni |n1 , · · · ni , · · · i
That is, |n1 , · · · ni , · · · i is an eigenvector of nbi , and its eigenvalue is the occupation number of the state i .
Occupation number operators are self-adjoint:
D E D E
n01 , · · · n0i , · · · abi † abi n1 , · · · ni , · · · = habi (n01 , · · · n0i , · · · ) |abi | n1 , · · · ni , · · · i = abi † abi (n01 , · · · n0i , · · · )|n1 , · · · ni , · · ·
and they commute among themselves, since, for i 6= j,
abi † abi abj † abj = abi † abj † abj abi = abj † abj abi † abi
Their eigenvalues univocally determine a complete set of state vectors, so that they are a complete set of
commuting observables. On the other hand they are idempotent (nbi 2 = nbi for the basis set {|n1 , · · · ni , · · · i}),
so that they are projection operators. nbi projects onto the the subspace spanned by all the Slater determinants
containing i .
The sum of occupation number operators for every spin-orbital is known as the electron number operator for
obvious reasons:
X
b=
n nbi
i
Second Quantization Formalism (2014-2018) 5
X
b |n1 , · · · ni , · · · i =
n ni |n1 , · · · ni , · · · i = n |n1 , · · · ni , · · · i
i
Different nbi do not project onto orthogonal subspaces, since nbi n cj |n1 , · · · ni , · · · i 6= 0 if ni = nj = 1, so
that theorem 7 cannot be applied5 and, in fact, their sum n b is not a projection operator. The restrictions of
the operators nbi to H1 do project onto orthogonal one-dimensional subspaces, so that the definition of n b is a
resolution of the identity in that subspace, and nb restricted to H1 is the identity operator in the spin-orbital
subspace.
In general, linear combinations of Slater determinants n —such as multiconfigurational wavefunctions—
are not eigenvectors of the occupation number operators, but their expected value can still be used to assign an
occupation number to each spin-orbital in the wave function, also referred to as the population of the spin-orbital:
* +
X X X
hni in = hn |nbi n i = CI n I |nbi CJ n J = CI⇤ CJ hn I |nbi n J i
I J IJ
Since nbi J =n
J if J contains i and vanishes otherwise, we can restrict the summation over J to the
n n
determinants containing that spin-orbital:
X X
hni in = CI⇤ CJ hn I |n J i = |CI |2 1
I,J3i I3i
Thus, spin-orbital populations are, in general, less than 1 for multiconfigurational wave functions. Certainly, all
of the n-electron determinants are eigenfunctions of the electron number operator n b with eigenvalue n, so that
the same applies to n .
3.2 Anticommutation rules
Electron creation and annihilation operators fulfill the following anticommutation rules:
h i h i
abi , abj † = ij [abi , abj ]+ = abi † , abj † = 0
+ +
h i
b B
where the anticommutator is defined as A, b bB
⌘A b+B
b A.
b
+
Let us prove the first rule. For i < j we have
⇣ ⌘
abi abj † + abj † abi |n1 , · · · ni , · · · i = abi ( 1)⌫j (1 nj ) |n1 , · · · 1 nj , · · · i + abj † ( 1)⌫i ni |n1 , · · · 1 ni , · · · i
⌫j ⌫i
= ( 1) (1 nj )( 1) ni |n1 , · · · 1 ni , · · · 1 nj , · · · i
⌫j0
+( 1)⌫i ni ( 1) (1 nj ) |n1 , · · · 1 ni , · · · 1 nj , · · · i = 0
since ⌫j0 = ⌫j ± 1, depending on ni being 0 or 1 respectively. For i = j
⇣ ⌘
abi abi † + abi † abi |n1 , · · · ni , · · · i = abi ( 1)⌫i (1 ni ) |n1 , · · · 1 ni , · · · i + ni |n1 , · · · ni , · · · i
= ( 1)2⌫i (1 ni )2 |n1 , · · · ni , · · · i + ni |n1 , · · · ni , · · · i = |n1 , · · · ni , · · · i
since ni or 1 ni must vanish.
According to these rules, if we commute a pair of annihilation or creation operators we have to introduce a
change of sign; that is, those pairs of operators anticommute:
abi abj = abj abi abi † abj † = abj † abi †
If the two operators are of either type then
abi abj † = ij abj † abi
so that they anticommute if they correspond to different states:
abi abj † = abj † abi for i 6= j
but abi does not anticommute (nor commute) with abi † :
abi abi † = 1 abi † abi
5A linear combination of projectors onto orthogonal subspaces is a projector if and only if all the coefficients are equal to 1.
6 Juan Carlos Paniagua
An immediate consequence of these rules is that we cannot create two electrons in the same state:
⇣ ⌘2 1 h † †i
abi † = abi , abi =0
2 +
in accordande with the fermionic character of these particles.
Exercise
Use the occupation-number representation of the Slater determinants to show that h ( i j ) | ( k l ) i =
ik jl il jk .
Hint: Write the determinants as creation operators acting on the vacuum state; then move the creation
operators from the left to the right-hand side of the scalar product; then move the resulting annihilation
operators to the right until they operate directly on the vacuum state.
4 The non-relativistic many-electron hamiltonian in second quanti-
zation
We will now obtain an expression of the non-relativistic hamiltonian operator of a many-electron system in
terms of creation and annihilation operators that is independent of the number of electrons in the system. This
makes it quite convenient for some mathematical developments and, in particular, for infinite systems such as
solids.
The non-relativistic electronic hamiltonian of a system with n electron and N nuclei is a sum of one-electron
and two-electron terms:
Xn n
X1 X n
n b 1
H= h(i) +
i=1
r
i=1 j=i+1 ij
r2i PN
where bh(i) = 2 A=1 riA .
ZA
We want to show that the second quantization formalism allows to put it in
the form
X 1X
b =
H hrs abr † abs + grstu abr † abs † a
cu abt
rs
2 rstu
D E D E
where hrs = r
b
h
s and grstu = r (1) s (2)
u (2) . To be precise, n H
1
t (1)
b is the projection of H
b
r12
onto the n-electron subspace of F, also referred to as the restriction of H b to that subspace. The sums extend
over the spin-orbitals of the one-electron basis { r }, there being no reference to n.
To prove the preceding statement we will show that n H b and H b (restricted to H⌦a n ) have the same matrix
1
elements for a given n-electron basis set. Let us first consider the one-electron part of H. b Its matrix element
are: * +
X †
X ⌦ ↵
( k 0 · · · l0 ) hrs abr abs ( k · · · l ) = hrs abr ( k0 · · · l0 ) |abs ( k · · · l )
rs rs
This vanishes unless r 2 {k 0 · · · l0 } and s 2 {k · · · l}. If both Slater determinants are equal the only non-vanishing
terms in the double sum correspond to r = s, and the corresponding diagonal matrix element reduces to
X ⌦ ↵ X
hrr abr ( k ··· l) |abr ( k ··· l) = hrr
r2{k···l} r2{k···l}
If the two basis vectors differ in one spin-orbital —say a in the former is replaced by b in the latter— then
the only surviving term in the double sum is the one with r = a and s = b; that is hab . If there are two
or more differing spin-orbitalsPn the matrix element vanishes. These results are the Slater-Condon rules for the
one-electron-type operator i=1 h(i).
Let us now consider the two-electron terms of the second quantized hamiltonian. For a diagonal matrix
element we have
* +
1X † † 1X ⌦ ↵
( k · · · l) cu abt ( k · · · l )
grstu abr abs a = grstu abs abr ( k · · · l ) |c
au abt ( k · · · l )
2 rstu 2 rstu
Second Quantization Formalism (2014-2018) 7
The terms in this sum vanish unless r = t 2 {k · · · l} and s = u 2 {k · · · l} or r = u 2 {k · · · l} and s = t 2
{k · · · l}. Thus, this diagonal element reduces
1 X ⌦ ↵ ⌦ ↵
grsrs abs abr ( k ··· l) |abs abr ( k ··· l) + grssr abs abr ( k ··· l) |abr abs ( k ··· l)
2
r,s2{k···l}
1 X X
= grsrs grssr = grsrs grssr
2
r,s2{k···l} r,s2{k···l},r>s
where we have taken into account the anticommutativity of the annihilation operators. Similar deductions
can be applied
Pn 1 for
Pnnon-diagonal matrix elements, leading to the Slater-Condon rules for the two-electron-type
operator i=1 j=i+1 r1ij . This completes the proof.
In the above demonstration we have assumed that the spin-orbitals r form a complete set. For computa-
tional reasons a finite subset must be used, so that the second quantized hamiltonian is, in fact, the projection
of the exact hamiltonian onto the subspace spanned by that subset.
Pn Pn 1 Pn
Similar expressions to those obtained for i=1 h(i) and i=1 j=i+1 r1ij can be used to write the second
quantized form of any one- or two-electron-type operator.
Exercise
Show that the second quantized form of the first-order reduced density operator b in an arbitrary or-
thonormal spin-orbital basis set is X †
b= rs abr abs
rs
and, for the natural spin-orbital basis,
X X
b= ni abi † abi = ni nbi
i i
Exercise
Use
Pn the anticommutation
P rules to show that a one-electron-type operator of an n-electron system, Fb =
†
i=1 f (i) = rs frs abr abs , can be cast into the form of a two-electron-type operator:
1 X 1 X
Fb = frt su abr
†
abs † a
cu abt = rt fsu abr
†
abs † a
cu abt
n 1 rstu n 1 rstu
both being restricted to the n-electron
P subspace of the Fock space. P Hint: use the anticommutation rules
to bring abt next to abr † to obtain rt frt abr † abt = Fb ; use also n cs .
b = sn
Use this result to write the n-electron hamiltonian as a sum of two-electron operators:
X 1 1
b =
H wrstu abr † abs † a
cu abt with wrstu = hrt su abr
†
abs † a
cu abt + grstu
rstu
n 1 2
The expected value of the hamiltonian for any n-electron state n
in in the second quantization formalism
is
D E X D E 1X D E
n b
H n
= hrs n
abr † abs n
+ grstu n abr † abs † acu abt n
rs
2 rstu
X 1X
n n
= hrs habr |abs | i+ grstu habs abr n |c
au abt | n i
rs
2 rstu
For the particular case of a single-determinant wave function n the only non-vanishing terms in the above
summations are those for which r, s, t and u correspond to spin-orbitals that are occupied in the determinant,
8 Juan Carlos Paniagua
and the usual Hartree-Fock-type energy expression is readily obtained:
D E occ
X 1X
occ
n b
H n
= hrs habr n
|abs | n
i+ grstu habs abr n
|c
au abt | n
i
rs
2 rstu
occ
X occ
1X
= hrr + grsrs grssr
r
2 rs
occ
X occ
X
= hrr + grsrs grssr
r r<s
Although the number of electrons does not appear explicitly in the above expressions, P it is implied in the lists
of occupation numbers of the n-electron basis vectors n I = |n1 , · · · ni , · · · i: n = i ni .
4.1 Restricted spin-orbitals
D E
Usually the spin-orbitals are chosen as products of an orbital r and a spin vector ↵ or . Then r b
r h s =0
D E
unless r and s have the same spin factor, and r s r12
1
t u = 0 unless r and t on the one hand, and
s and u on the other, have the same spin factor. By carrying out the scalar products of the spin factors we are
left with scalar products involving only orbitals. Thus, for a closed-shell determinant the electronic hamiltonian
takes the form: X X
b = † 1X X † †
H hrs adr! a d
s! + grstu ad
r! ac
s⌧ adu⌧ ac
t!
rs
2 rstu
!=↵, !,⌧ =↵,
D E D E
where hrs = b
r h s , grstu =
1
r (1) s (2) r12 t (1) u (2) and the indexes r, s, t and u extend over the
orbital basis. As told before, this basis set must be truncated in practice to a finite number m, so that we work
on a 2m-dimensional subspace of H1 . Then the sums over r, s, t and u in the preceding equation extend over
those m orbitals and the resulting second-quantized operator is an approximation to the true hamiltonian n H. b
5 Change of spin-orbital basis set
For each spin-orbital basis set there is a corresponding set of creation and annihilation operators, and different
†
sets can be connected by using the appropriate resolutions of the identity. So, the operators ab0r corresponding
to a new basis set { 10 , · · · r0 , · · · } should satisfy the equation
!
† X X
b 0
ar |i = r = 0
| ii h i| 0
r = h i | r0 i abi † |i
i i
which is clearly fulfilled if
† X
ab0r = h i|
0
bi †
ri a
i
By applying his relationship to each basis vector of the Fock space it can be shown to be general. This
relationship is, in fact, the same that connects the elements of the two basis sets:
X
0
r = h i | r0 i i
i
By taking adjoints in both sides of the transformation equation for creation operators one obtains the corre-
sponding relation for the annihilation operators:
X
ab0r = h r0 | i i abi
i
5.1 Field operators
If we change to a continuous basis set, as the position eigenvectors {|~r, !i}, the resulting creation and annihilation
operators will be functions of a continuous index (~r), and they are called field operators or quantum field
operators. The preceding equation then becomes
X X
b
a(~r, !) = h~r, !| i i abi = r, !)abi
i (~
i i
Second Quantization Formalism (2014-2018) 9
and, by taking adjoints, X
a† (~r, !) =
b ⇤
r, !)abi †
i (~
i
a (~r, !) creates a particle at point ~r with spin !:
b†
X X
a† (~r, !) |i =
b h i | ~r, !i abi † |i = | ii h i| ~
r, !i = |~r, !i
i i
The expression abi of in terms of the field operator ba(~r, !) can be readily obtained from the expression of this
operator in terms of the abj ’s:
X ˆ X ˆ X X
⇤ ⇤
i (~
r , !)b
a (~
r , !)d~
r = i (~
r, !) r, !)abj d~r =
j (~ ij abj = a
bi
!=↵, ~
r !=↵, ~
r j j
By taking adjoints in both sides of this equation the corresponding expression for abi † is obtained:
X ˆ
i (~ a† (~r, !)d~r = abi †
r, !)b
!=↵, ~
r
Field operators fulfill similar anticommutation relationships than discrete creation and annihilation opera-
tors:
2 3
⇥ ⇤ X X X h i
†5 †
b a† (~r0 , ! 0 ) + = 4
a(~r, !), b i (~
r , !) b
a i , ⇤ 0
j (~
r , ! 0
) b
a j = i (~
r , !) ⇤ 0
j (~
r , ! 0
) b
a ,
i jb
a
+
i j i,j
+
X X
= r, !) j⇤ (~r0 , ! 0 ) ij
i (~ = h~r, !| ii h i| ~
r 0
, ! 0 i = h~r, !| ~r0 , ! 0 i
i,j i
that, is ⇥ ⇤
b a† (~r0 , ! 0 ) + =
a(~r, !), b !! 0 (~r, ~r0 )
In the same way X
a(~r0 , ! 0 )]+ =
a(~r, !), b
[b i (~
r, !) j (~r 0
, ! 0 ) [abi , abj ]+ = 0
i,j
and, by taking adjoints: ⇥ † ⇤
b a† (~r0 , ! 0 ) + = 0
a (~r, !), b
Field operators are usually denoted by b(~r, !) and b† (~r, !) . Many texts omit the accent ^ in the notation
of operators, which can lead to confusion with the notation normally used for wave functions.
6 Particles and holes
Electron creation and annihilation operators are sometimes referred to a Fermi vacuum or Fermi sea instead
of the zero-electron vacuum. The Fermi sea is the independent-electron ground state, in which all the electrons
occupy the lowest-energy spin-orbitals. The energy of the highest occupied spin-orbital is known as the Fermi
level. The independent-particle excited states are identified by specifying their occupation number differences
with respect to the ground state vector; that is, the holes created in the Fermi sea by annihilation operators
and the particles created above the Fermi level by creation operators. That is, an operator that annihilates an
electron below the Fermi level is viewed as a hole creation operator. A hole acts as a particle with positive charge
e (a quasi-particle), and a neutral pair formed by an electron and a hole interacting by electrostatic attraction
is sometimes called an exciton. The Fermi sea can be considered as a new vacuum with no particles (above the
Fermi level) and no holes (below the Fermi level).
The language of particles and holes is common in solid-state theory, and it is also sometimes used for finite
systems, particularly in the statement of post-Hartree-Fock methods.
7 Non-fixed-particle systems
It is clear that the second quantized operator of any observable in fixed-particle quantum mechanics must contain
an equal number of creation and annihilation operators, so that these should always appear in pairs of either
type. However, single operators that create or annihilate photons are needed to study spectroscopic phenomena
10 Juan Carlos Paniagua
in which de quantum nature of light plays a relevant role, such as the spontaneous emission of radiation or the
Raman scattering. Since photons have spin 1 they are bosons and the corresponding creation and annihilation
operators are defined otherwise (see, for instance, Quantum electrodynamics by José A. N. F. Gomes and Juan
C. Paniagua, in Computational Chemistry: Structure, Interactions and Reactivity, ed. by S. Fraga. Studies in
Physical and Theoretical Chemistry, vol. 77 (B). Elsevier, Amsterdam (1992)).