Download Chapter 12 Quantum gases

Chapter 12
Quantum gases
In classical statistical mechanics, we dealt with an ideal gas which was a good approximation for a real gas in the highly diluted limit.
An important difference between classical and quantum mechanical many-body systems
lies in the distinguishable character of their constituent particles. The inherent indistinguishability in quantum mechanics gives rise to phenomena and concepts that are not
present in classical physics.
In this chapter, we will present the ideal quantum gases, the Fermi gas and the Bose gas,
as well as some practical examples of these systems.
12.1
Identical particles
We consider a system of N non-interacting particles:
Ĥ =
N
X
Ĥ (i) ,
i=1
with
(i) Ĥ (i) ϕ(i)
αi = εα i ϕ α i .
(12.1)
E
(i)
In eq. (12.1), ϕαi are the one-particle wavefunctions, with the subindex αi denoting a
complete set of quantum numbers such as, for instance, (n, l, ml , ms ), (kx , ky , kz , ms ), etc.
In the case of N distinguishable particles, we can enumerate them so that the wavefunction
of the N -particle system is given as a product of the one-particle states:
(2) (N ) ϕα . . . ϕα .
|ϕN i ≡ |ϕα1 , . . . , ϕαN i ≡ ϕ(1)
α1
2
N
If the particles are indistinguishable, then the principle of indistinguishability imposes
certain symmetry properties on the N -particle state, i.e., that
⋆ the interchange of two particles results in, at most, the change of the sign of the
N -particle wavefunction.
153
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CHAPTER 12. QUANTUM GASES
Therefore, the N -particle wavefunction has to be properly symmetrized:
E
(±)
≡ |ϕα1 , . . . , ϕαN i(±)
ϕN
(2) (N ) 1 X
ϕα . . . ϕ α ) .
(±)p P(ϕ(1)
≡
α1
2
N
N! P
p: number of pair interchanges (transpositions) performed by the permutation P.
E
(+)
or antisymThe state of a system of identical particles can be either symmetrical ϕN
E
(−)
metrical ϕN . These two types of states are orthogonal to one another; they belong to
(+)
(−)
different Hilbert spaces HN and HN .
Pauli spin-statistics theorem
Wolfgang Pauli showed that, for a system of indistinguishable particle of certain sort,
the spin of the particles and the nature of the system’s wavefunction (symmetrical or
antisymmetrical) are related to one another.
E
(+)
(+)
HN : Hilbert space of symmetrical states ϕN
of a system that consists of identical
particles with integer spin (s = 0, 1, 2, 3, . . .) called bosons .
Examples: photons (s = 1), phonons (s = 1), magnons (s = 1), α-particles
(s = 0).
E
(−)
(−)
HN : Hilbert space of antisymmetrical states ϕN of a system that consists of identical
particles with half-integer spin (s = 12 , 23 , . . .) called fermions .
Examples: electrons (s = 1/2), protons (s = 1/2), neutrons (s = 1/2).
The N -particle wavefunction of a fermionic
minant:
(1) E
ϕ α1
E
1 (−)
..
ϕN =
N! . E
ϕ(1)
αN
system can be represented by a Slater deter
E
(2)
...
ϕ α1
..
. E
(2)
...
ϕαN
E
(N )
ϕ α1
..
. E
(N )
ϕαN
.
E
(−)
If two raws or two columns of the determinant are identical then ϕN
= 0. This is a
consequence of the Pauli principle:
”two fermions can never coincide in all of their quantum numbers”
155
12.2. FOCK SPACE
12.2
Fock space
A very different way of representing the N -particle wavefunction for a system of fermions
or bosons is by considering the occupation numbers nαi .
E
(±)
nαi : number of times that the one-particle state |ϕαi i is present in ϕN .
In terms of nαi , the N -particle state is written as





X
(p) (p+1) 
(±)
(1) (2)
p
(±) P ϕα1 ϕα1 . . . ϕαi ϕαi
|N ; nα1 . . . nαi . . .i ≡ c±
... ,

{z
} |
{z
} 
|

P
n α1
n αi
with
N =
X
n αi
and
i
c± = p
N!
1
Q
i
n αi !
.
(12.2)
c± is calculated from the normalization for the orthogonal Fock states:
Y
(±)
δnαi ñαi .
N ; . . . nαi . . . |Ñ ; . . . ñαi . . . (±) = δN Ñ
i
Note that
nαi = 0 or 1
for fermions,
nαi = 0, 1, 2, . . . for bosons.
(±)
The Fock states constitute a complete orthonormal basis in HN .
Second quantization
Having extended our description of a quantum system by considering the Fock space, we
can introduce now the formalism of second quantization.
We define the creation (annihilation) operators a†αi and aαi as
|ϕαi i = a†αi |0i ,
aαi |ϕαi i = |0i ,
aαi |0i = 0,
|0i: vacuum state.
The operator a†αi (aαi ) creates (annihilates) the one-particle state |ϕαi i. By considering
the corresponding normalization factors (QM II), these operators are defined in terms of
the N -particle wavefunction as follows:
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CHAPTER 12. QUANTUM GASES
Bosons:
p
nαr + 1 |N + 1; . . . nαr + 1 . . .i(+) ,
√
nαr |N − 1; . . . nαr − 1 . . .i(+) ,
=
a†αr |N ; . . . nαr . . .i(+) =
aαr |N ; . . . nαr . . .i(+)
nαr = 0, 1, 2, . . . .
Fermions:
a†αr |N ; . . . nαr . . .i(−) = (−1)Nr δnαr ,0 |N + 1; . . . nαr + 1 . . .i(−) ,
aαr |N ; . . . nαr . . .i(−) = (−1)Nr δnαr ,1 |N − 1; . . . nαr − 1 . . .i(−) ,
nαr = 0, 1;
Nr =
r−1
X
n αj .
j=1
The Fock states can be represented by the creation operators as
n j
a†αj
Y
p
|N ; . . . nαr . . .i(±) =
(±1)Nj |0i .
nj !
j
In this representation, we get rid of the symmetrization/antisymmetrization aspect of the
wavefunction. The information about the symmetry of the wavefunction is now contained
in the commutation relations of the creation/annihilation operators:
• for bosons,
• for fermions,
[aαr , aαs ] = a†αr , a†αs = 0,
aαr , a†αs = δrs ;
{aαr , aαs } = {a†αr , a†αs } = 0,
{aαr , a†αs } = δrs .
Above, [ , ] denotes a commutator and { , } denotes an anticommutator. Usually, one
uses a and a† for denoting fermionic creation/annihilation operators and b and b† for
bosonic ones.
Quantum mechanical operators describing observables can also be represented in terms
of creation and annihilation operators.
- Occupation number operator:
n̂αr = a†αr aαr .
12.3. PARTITION FUNCTION OF THE IDEAL QUANTUM GASES
157
- Particle number operator :
N̂ =
X
n̂αr =
r
X
a†αr aαr .
r
- For an ideal quantum gas of N independent particles, the Hamiltonian operator is
given by
N
X
(i)
Ĥ =
Ĥ1 ,
i=1
with
(i)
Ĥ1 =
1 2
p̂ + V (r̂i ).
2m i
V (r̂i ) describes the possible interaction of particle i with an external potential V
(electric field, magnetic field, etc). In second quantization, Ĥ can be written as
X
X
Ĥ =
εr a†αr aαr =
εr n̂αr ,
r
r
with
εr δrs = hεr | Ĥ1 |εs i .
The Fock states are the eigenstates of Ĥ:
Ĥ |N ; . . . nαr . . .i
12.3
(±)
=
X
εr n α r
r
!
|N ; . . . nαr . . .i(±) .
Partition function of the ideal quantum gases
The most adequate ensemble to deal with quantum gases is the grand canonical ensemble:
Z(T, V, µ) = Tr e−β(Ĥ−µN̂ ) ,
(12.3)
Ĥ: Hamilton operator,
N̂ : particle number operator.
In order to evaluate the trace, it is very useful to consider the Fock states, which are
eigenstates of both the particle number operator and the Hamilton operator. This allows
us to rewrite (12.3) as
Z
(±)
(T, V, µ) =
∞
X
N =0
=
∞
X
N =0
P
P
nr =N
X
r
e−β
P
r
nr (εr −µ)
{nr }
nr =N
X
Y
r
{nr }
r
e−βnr (εr −µ) ,
(12.4)
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CHAPTER 12. QUANTUM GASES
where Z (+) (T, V, µ) is the partition function for an ideal gas of bosons and
Z (−) (T, V, µ) is the partition function for an ideal gas of fermions,
and nr was used instead of nαr for brevity.
The sum {nr } in eq. (12.4) runs over all combinations of occupation
numbers for a given
P
total number of particles N . In fact, since the summation ∞
frees
the constraint of
N =0
P
nr =N
X
r
,
{nr }
one can alternatively write the double sum
∞
X
N =0
as
XX
n1
P
nr =N
X
r
(12.5)
{nr }
...
n2
X
...,
(12.6)
nr
P
where the sums nr are independent. Now, using the equivalence between the summations (12.5) and (12.6), we calculate the partition function:
!
!
!
X
X
X
e−βnr (εr −µ)
e−βn1 (ε1 −µ)
e−βn2 (ε2 −µ) . . .
Z (±) (T, V, µ) =
n2
n1
=
Y X
r
nr
e−βnr (εr −µ)
!
.
nr
(12.7)
Eq. (12.7) can be further simplified if one makes use of the statistical properties of
particles in a given system. Thus,
• for a system of bosons , nr runs over all non-negative integers so that in the
parenthesis of eq. (12.7) we have a geometrical series:
Y
1
(+)
Z (T, V, µ) =
,
1 − e−β(εr −µ)
r
• for a system of fermions , nr = 0, 1 and therefore we only have two terms in eq.
(12.7):
Y
Z (−) (T, V, µ) =
1 + e−β(εr −µ) .
r
Out of Z (±) one can obtain all the thermodynamical properties of the ideal quantum
gases.
12.4. THERMODYNAMICAL PROPERTIES OF IDEAL QUANTUM GASES
12.4
159
Thermodynamical properties of ideal quantum
gases
(i) Grand canonical potential :
Ω(+) (T, V, µ) = −kB T ln Z (+) (T, V, µ)
X = kB T
ln 1 − e−β(εr −µ) ;
r
Ω(−) (T, V, µ) = −kB T ln Z (−) (T, V, µ)
X = −kB T
ln 1 + e−β(εr −µ) .
r
⋆ Note that the volume dependence of Ω(±) (T, V, µ) is in the one-particle energies
εr . Thus, for instance, if the particles are in a potential well with infinitely
high walls, then, accordingly, ǫr will have only certain allowed values.
(ii) Expectation value of the particle number operator N̂ :
X
1
1 ∂
ln Z (+) (T, V, µ) =
;
β(ǫr −µ) − 1
β ∂µ
e
r
X
1 ∂
1
(−)
=
ln Z (T, V, µ) =
.
β(εr −µ) + 1
β ∂µ
e
r
< N̂ >(+) =
< N̂ >(−)
Out of these equations, we can obtain the chemical potential
µ = µ(T, V, < N̂ >(±) ).
(iii) Thermal equation of state for the ideal quantum gas:
P V = kB T ln Z (±) (T, V, µ).
(iv) Internal energy for bosons:
∂
ln Z (+) (T, V, µ) + µ < N̂ >(+)
∂β
X εr − µ
+ µ < N̂ >(+)
=
β(εr −µ) − 1
e
r
X
εr
=
β(ε
−µ)
r
e
−1
r
U (+) = −
and for fermions:
U (−) =
X
r
εr
.
eβ(εr −µ) + 1
(12.8)
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CHAPTER 12. QUANTUM GASES
12.5
Particle distribution function for bosons and fermions
We define the average occupation number < n̂r >(±) of the rth one-particle state,
P
< n̂r >(±)
nl =N
X lX
P
1
[−β l nl (εl −µ)]
=
n
e
r
Z (±) (T, V, µ) N
{nl }
=
−
1 ∂
ln Z (±) (T, V, µ) ,
β ∂εr
as the distribution function of particles in the state r. This quantity is especially important
for understanding quantum gases.
For bosons and fermions we have, correspondingly,
→ Bose-Einstein distribution function:
1
< n̂r >(+) =
eβ(εr −µ)
−1
= f (+) (εr )
and
→ Fermi-Dirac distribution function:
< n̂r >(−) =
1
eβ(εr −µ)
+1
= f (−) (εr ).
Properties:
(1) For fermions, µ can have any value and
0 ≤ < n̂r >(−) ≤ 1
(Pauli principle).
(2) For bosons, µ must be smaller than the smallest one-particle energy ε0 , otherwise
the occupation number < n̂0 > would be negative and diverge for ε0 = µ. For
T → 0, this would mean that all occupation numbers are zero and therefore the
total number of bosons N → 0. This would be fine for the case that we have no
bosons. But what happens when N is given and nonzero?
⋆ In order to treat the limit T → 0, we make an assumption that in this limit
µ → ε0 in a way that the lowest one-particle state is macroscopically occupied,
i.e.,
< n̂0 >(+) (T = 0) = N
⇒
Bose-Einstein condensation.
12.5. PARTICLE DISTRIBUTION FUNCTION FOR BOSONS AND FERMIONS 161
(3) Making use of < nr >(±) , we can calculate the average total number of particles and
the internal energy as
< N̂ >(±) =
X
< nr >(±) ,
r
U
(±)
=
X
εr < n̂r >(±) .
r
(4) For large one-particle energies, i.e., for εr − µ >> kB T , both Bose-Einstein and
Fermi-Dirac distribution functions converge to the Maxwell-Boltzmann distribution
function:
(εr − µ >> kB T ),
< n̂r >(±) ∼ e−βεr
which implies that in the classical limit there is no difference between bosons and
fermions. The meaning of this limit will be explicitly discussed in the next chapter
Graphical representation of the distribution functions
We define x = β(εr − µ) and plot < nr > vs. x for the three distribution functions (Fig.
12.1).
Figure 12.1: Fermi-Dirac (F-D), Bose-Einstein (B-E) and Maxwell-Boltzmann (M-B) distribution functions vs. x.
From Fig. 12.1, the limit x >> 1, corresponding to εr − µ >> kB T , where all the
distribution functions coincide, is the classical limit. For x → 0, the distribution function
for bosons diverges in order to avoid the situation when µ > ε, as mentioned previously.
In the next graphic Fig. 12.2, we explicitly show the distribution function for fermions.
For T = 0, all states up to εr = µ are occupied (with one particle each (Pauli principle))
and all states with εr > µ are empty. For T = 0, the chemical potential coincides with
the Fermi energy: µ = εF .
For T > 0, fermions get statistically excited to higher energy states and their distribution
function is as shown in Fig. 12.2.
Exercise: find the particle fluctuations for bosons, fermions and classical particles.
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CHAPTER 12. QUANTUM GASES
Figure 12.2: Fermionic distribution function vs. temperature T .
The relative particle fluctuations with respect to the average occupation number is
given by:
σn2 r
1
< n̂2r > − < n̂r >2
=
= eβ(εr −µ) = z −1 eβεr =
−c,
2
2
< nr >
< nr >
< nr >
where z is the fugacity and the constant c is
cM −B = 0
cB−E = −1
cF −D = +1
for classical particles (Maxwell-Boltzmann),
for bosons
(Bose-Einstein),
for fermions
(Fermi-Dirac).
The particle fluctuations for bosons are larger than those for classical particles while
for fermions they are smaller. The reason for that is that for fermions the change
of a particle state is hindered by the Pauli principle while it is enhanced for bosons!