The second quantization method for indistinguishable particles (Lecture Notes in Physics, UFABC ...

59 Pages·2013·0.4 MB·English

arXiv:1308.3275v1 [cond-mat.quant-gas] 14 Aug 2013
The second quantization method
for indistinguishable particles
Valery Shchesnovich
Lecture Notes for postgraduates
UFABC, 2010
“All I saw hast kept safe in a written record, here thy worth and
eminent endowments come to proof.”
Dante Alighieri, Inferno
Contents
1 Creation and annihilation operators for the system of indistin-
guishable particles 4
1.1 The permutation group and the states of a system of indistin-
guishable particles . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Dimension of the Hilbert space of a system of indistinguishable
particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Definitionandpropertiesofthecreationandannihilationoperators 9
1.4 The Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 The representations of state vectors and operators . . . . . . . . 16
1.5.1 N-particle wave-functions . . . . . . . . . . . . . . . . . . 16
1.5.2 The second-quantizationrepresentation of operators . . . 18
1.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Evolution of operators in the Heisenberg picture . . . . . . . . . 22
1.7 Statistical operators of indistinguishable particles . . . . . . . . . 24
1.7.1 The averagesof the s-particle operators . . . . . . . . . . 27
1.7.2 Thegeneralstructureoftheone-particlestatisticaloperator 29
2 Quadratic Hamiltonian and the diagonalization 31
2.1 Diagionalizationofthe Hamiltonianquadraticinthe fermionop-
erators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Diagionalization of the Hamiltonian quadratic in the boson op-
erators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 DiagonalizationofthequadraticbosonicHamiltonianpos-
sessing a zero mode . . . . . . . . . . . . . . . . . . . . . 41
2.3 Long range order,condensation and the Bogoliubov spectrum of
weakly interacting Bose gas . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 The excitation spectrum of weakly interacting Bose gas
in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 The excitation spectrum of an interacting Bose gas in an
external potential. . . . . . . . . . . . . . . . . . . . . . . 50
2.4 TheJordan-Wignertransformation: fermionizationofinteracting
1D spin chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3
Chapter 1
Creation and annihilation
operators for the system of
indistinguishable particles
The properties of the permutation group are reviewed. The projectors on the
symmetric and anti-symmetric subspaces of the Hilbert space of the system of
identical particles are considered. The dimension of the physical Hilbert space
of a system of indistinguishable identical particles is calculated. The creation
andannihilationoperatorsareintroducedstartingfromthestatestheycreateor
annihilate. Thepropertiesofthecreationandannihilationoperatorsarederived
from their definition. The basis of the physicalHilbert space is expressedusing
the creation operators applied to the vacuum state. The representation of the
arbitrary vectors in the physical Hilbert space is given in terms of these basis
states. Itisshownhowtoexpandthes-particleobservablesintheoperatorbasis
composed of the creation and annihilation operators, which corresponds to the
“bra-ket” vector basis of the observables describing distinguishable particles.
The evolution equation for the creation and annihilation operators is derived.
The statistical operators are also introduced and related to the average of the
observables. Some examples are provided.
4
1.1 The permutation group and the states of a
system of indistinguishable particles
Wewillsaythattheparticlesareidenticalifthey havethesameproperties. On
the other hand, the particles are considered indistinguishable if they cannot be
labelled with an index.
The Hilbert space of a system consisting of N identical (distinguishable!)
particles is constructed from the tensor product of the states describing each
particlealone,for instance ifN particles arefoundin the states labelledas X ,
α
X , ..., X we have
β γ
X(1),X(2),...,X(N) X X ... X . (1.1)
| α β γ i≡| αi| βi· ·| γi
Here and below we will use the upper index for labelling of the particles, while
inthetensorproduct(ther.h.s. ofEq. (1.1))thepositionofthevectoridentifies
to which particle it corresponds. The arbitrary vector of the Hilbert space of
N identical particles is then given by an arbitrary linear combination of the
vectors defined in Eq. (1.1).
A permutation P of N ordered elements (X ,X ,...,X ) is an operation
1 2 N
which changes the order of the elements. A good way to visualize it is to
think thatthe elementsareplacedinthe stringofboxeslabelledby the natural
numbersandthe operationP swapsthe contentsofthe boxes. Forinstance,we
can write
P
(X ,X ,...,X ) (X ,X ,...,X ), (1.2)
1 2 N −→ i1 i2 iN
i.e. implying that i k or P(i ) = k. In other words, under the action
k k
→
of P of Eq. (1.2) the content from the i -th box goes to the box labelled
k
with the index k (while the content of the k-th box goes somewhere else). The
permutationoperationsformagroup,sincethecompositionoftwopermutations
is also a permutation and each permutation has the inverse one. The trivial
permutation, which leaves the order unchanged, will be defined by I. The
simplest permutationoperationis the transposition,i.e. the interchangeofjust
two elements from the ordered set, for example for i = j we define P (i) = j
ij
6
and P (j) = i with the rest of the boxes preserving their content under P .
ij ij
This permutation will be denoted below as simply (i,j). Note that it is inverse
to itself (i,j)(i,j)=I. The following useful property
(i,j)=(i,1)(j,2)(1,2)(i,1)(j,2) (1.3)
can be verified by simple calculation. Moreover, it is easy to see that an ar-
bitrary permutation can be represented as composition of the transpositions
(moreover,any permutation can be written as a composition of the elementary
transpositions of the form (i,i+1)). Finally, there are N! of all permutations
of a set of N elements.
The signatureofapermutation, denotedasε(P), isdefinedasa mappingof
the permutation group to the group of two elements 1,1 , where the usual
{− }
productis the groupoperationonthe latterset. UsingEq. (1.3)andthe group
5
property one can establish that there are just two possible ways to attribute a
signature to the permutation: the trivial one, i.e. P 1 and the one which
→
attributes the signature 1 to the permutation (1,2) and, by Eq. (1.3), to any
−
permutation being a transposition, ε(P ) = 1 for i = j (in this case, the
ij
− 6
signature is called the parity). To verify that it is indeed possible to define the
parity of P (as preservingthe groupoperationmapping to the set 1,1 )one
{− }
canproduce its explicit expression,givenas aproduct ofthe partialsignatures,
i.e.
ε(P) sgn[P(j) P(i)], (1.4)
≡ −
i<j
Y
where sgn is the sign function and the permutation P acts on the ordered
set (1,2,...,N), e.g. as in Eq. (1.2). First, to verify the group property
consider the composition of two permutations P and P , i.e. the permutation
2 1
P (k)=P (P (k)) or P =P P . We have
3 2 1 3 2 1
◦
ε(P P )= sgn[P (P (j)) P (P (i))]
2 1 2 1 2 1
◦ −
i<j
Y
= sgn[P (j) P (i)] sgn[P (P (j)) P (P (i))]
1 1 2 1 2 1
− −
iY<j P1(i)Y<P1(j)
=ε(P )ε(P ).
1 2
Second,obviouslyε(P )= 1. Itisclearthattheparityε(P)=( 1)s,wheres
12
− −
isthe numberofthetranspositionsinarepresentationofP (asabyproduct,we
getthat anytwosuchrepresentationsmusthavethe sameparityofthe number
of transpositions).
LetusnowconsiderthepermutationoperationasactingontheHilbertspace
of N identical particles. Since it is a linear space, the permutation operation
P now becomes a linear operator (for which we will use the same notation P).
Namely,wedefinetheactionoftheoperatorP correspondingtothepermutation
in Eq. (1.2) as follows
P X(1),X(2),...,X(N) X(1),X(2),...,X(N) , (1.5)
| 1 2 N i≡| i1 i2 iN i
or in the product form
P X X ... X = X X ... X .
| 1i| 2i· ·| Ni | i1i| i2i· ·| iNi
It is clear from this definition that the permutation operator is unitary, i.e.
P =P 1.
† −
If the particles are considered as indistinguishable, the interchange of any
twomust notaffectthe state ofthe system(exceptfor a constantphase). Thus
if S isthestateofsuchasystem,wemusthaveP S =eiϕ(P) S ,wherethe
| i | i | i
phasedepends onthe permutationP. Onthe otherhand,foranytransposition
P = (1,2) we must have P2 S = S , since P2 = I. Thus ϕ(P ) = 0 or π
ij ij | i | i ij ij
and,duetoEq. (1.3),itisthesame phaseforanytransposition. Inotherwords,
6
one should use the state which are either symmetric or anti-symmetric in the
interchangeoftwoparticles. Theparticleswhicharedescribedbythesymmetric
statesarecalledbosons andtheonesdescribedbytheanti-symmetricstatesare
called fermions. The states which satisfy the property P S = ε(P) S , i.e.
| i | i
symmetric and anti-symmetric states, will be called the physical states of the
indistinguishable particles, or simply the physical states.
To construct a physical state from a product state of N identical particles
one can use the projector operators S and A defined as follows
N N
1 1
S P, A ε(P)P, (1.6)
N N
≡ N! ≡ N!
P P
X X
wherethesummationrunsoveralltranspositionsofthesetofN elements(note
that in the sum we have operators, while in the index of the summation the
corresponding transpositions). The projector property is readily verified:
2 2
1 1
S2 = P P = PP
N N! ′ N! ′
(cid:18) (cid:19) P P′ (cid:18) (cid:19) P P P′
X X X X◦
1
= PP =S
′ N
N!
P P′
X◦
and
2 2
1 1
A2N = N! ε(P)P ε(P′)P′ = N! ε(P)ε(P′)PP′
(cid:18) (cid:19) P P′ (cid:18) (cid:19) P P P′
X X X X◦
1
= ε(PP )PP =A .
′ ′ N
N!
P P′
X◦
Moreover,thetwoprojectorsinEq. (1.6)areorthogonal,i.e. thesymmetricand
anti-symmetric subspaces are orthogonal(which is a necessary property if they
are to describe two different types of particles, bosons and fermions). Indeed,
by similar calculation one gets
2 2
1 1
S A = P ε(P )P = ε(P) ε(P)ε(P )PP
N N ′ ′ ′ ′
N! N!
(cid:18) (cid:19) P P′ (cid:18) (cid:19) P P P′
X X X X◦
1
= ε(P) A =0,
N
N!
!
P
X
sinceonecandivideallthepermutationoperatorsintotwoclassesbymultiplica-
tion by P : P˜ =P P, where the operators relatedby P have the signatures
12 12 12
of different sign.
The projectors S and A are also Hermitian, which is a consequence of
N N
the unitarity ofthe permutation operators,P =P 1 and the propertyε(P)=
† −
7
ε(P 1). It is easy to see that only in the case of just two particles the sum of
−
these projectors is the identity operator, i.e. S +A =I .
2 2 2
To unify the consideration, we will use the same notation ε(P) for the sig-
nature in the boson and fermion cases. Thus we will write the “generalized
symmetrization” as
1
Sε = ε(P)P, (1.7)
N N!
P
X
where ε(P)=1 for the case of S .
N
The states of N indistinguishable particles can be given as linear combina-
tions of the following states
X ,X ,...,X =Sε X(1),X(2),...,X(N)
| 1 2 Ni N{| 1 2 N i}
=Sε X X ... X , (1.8)
N{| 1i| 2i· ·| Ni}
where (and below) we use the notation X ,X ,...,X for a symmetric or
1 2 N
| i
anti-symmetricstate ofN particles occupyingthe single-particlestates labelled
by X , X , ..., X (not necessarily different). Explicitly, we have (compare
1 2 N
with Eq. (1.2))
1
|X1,X2,...,XNi= N! ε(P)|XP−1(1)i|XP−1(2)i·...·|XP−1(N)i (1.9)
P
X
Note that this vector is not normalized, in general. If two single-particle states
are identical, X = X then the corresponding anti-symmetric state is absent
i j
(one obtains zero on the r.h.s. of Eq. (1.9)) – this is nothing but the Pauli
exclusion principle for fermions.
1.2 Dimension of the Hilbert space of a system
of indistinguishable particles
In the previoussectionwe haveconsideredthe states of a system ofN indistin-
guishable particles. Consider now the Hilbert space of such states, the physical
Hilbert space for below. It is a subspace of the product of N identical single-
particleHilbert spaces, H H H ... H. We willdenote the Hilbert space
⊗ ⊗ ⊗ ⊗
ofN indistinguishableparticlesas Sε H H H ... H . Assuming
HN ≡ N{ ⊗ ⊗ ⊗ ⊗ }
that the dimension of H be d, i.e. dim(H) = d, let us find the dimension of
the Hilbert space . The dimension of the Hilbert space (B) for bosons,
HN HN
occupying d single-particle states is the total amount of various distributions
of the N indistinguishable particles between the d states, such that there are
exactly m particles in the j-th state, i.e. (using the usual δ-symbol)
j
N N N N 2π
dim(HN(B))= ... δPdj=1mj,N = ... 2dπθeiθ(Pdj=1mj−N)
mX1=0 mXd=0 mX1=0 mXd=0Z0
8
2π 2π d
N N d N
dθ dθ
= ... e−iNθ eimjθ = e−iNθ eimθ
2π 2π
!
mX1=0 mXd=0Z0 jY=1 Z0 mX=0
2πdθ 1 ei(N+1)θ d
= e−iNθ − .
2π 1 eiθ
Z (cid:18) − (cid:19)
0
Now we change the variable to z = eiθ and use the theorem of residues to
calculate the integral (there is a pole at z = 0, while at z = 1 the integrand is
regular)
1 1 zN+1 d 1 dN 1 zN+1 d
dim(HN(B))= 2iπ dzz−N−1 −1 z = N!dzN −1 z (cid:12)
zZ=1 (cid:18) − (cid:19) (cid:18) − (cid:19) (cid:12)z=0
| | (cid:12)
(cid:12)
1 (cid:12)
= d(d+1) ... (d+N 1).
N! · · −
Therefore, for bosons we have obtained
1
dim( (B))= d(d+1) ... (d+N 1)=CN , (1.10)
HN N! · · − N+d−1
where the symbol Cn is defined as Cn m! .
m m ≡ (m n)!n!
On the other hand, in the case of fermion−s by applying the Pauli exclusion
principle (and the indistinguishability of the particles), we get
0, N >d
dim( (F))= =CN. (1.11)
HN 1 d(d 1) ... (d N +1), N d d
(cid:26) N! − · · − ≤
1.3 Definition and properties of the creation and
annihilation operators
The creation a+β and annihilation a−β operators, which “create”/“annihilate”1
the single-particle state β, while acting on the state of N indistinguishable
particles, are defined by the following rules
a−β |α1,α2,...,αNi=√Nhβ(1) |α1,α2,...,αNi, (1.12)
a+ α ,α ,...,α =√N +1Sε β α ,α ,...,α , (1.13)
β | 1 2 Ni N+1{| i| 1 2 Ni}
where the state α ,α ,...,α is the N-particle state defined as in Eq. (1.9).
1 2 N
| i
Thus, in Eq. (1.12) the product is understood as the scalar product in the
single-particleHilbertspacewiththeindex i=1whenthe stateofthe indistin-
guishable particles is written as a linear combination of the states of identical
particles, Eq. (1.9). Similar, the action of the projector Sε is defined by
N+1
1ThisisexactlywhattheiractionmeansintheFockspace,seebelow.
9