VII.B Canonical Formulation
Using the states constructed in the previous section, we can calculate the canonical
density matrix for non-interacting identical particles. In the coordinate representation we
have
�{~x ′ }|ρ|{~x}�η =
′ X
X
{�
k} P,P
′
1
,
η P η P �{~x ′ }|P ′ {~k}�ρ({~k})�P {~k}|{~x}�
Nη
′
(VII.11)
h
P
i
P′
N
2 2
where ρ({~k}) = exp −β
h̄
k
/2m
/ZN . The sum, {�k1 ,k� 2 ,···,k� N } , is restricted to
α
α=1
ensure that each identical particle state appears once and only once. In both the bosonic
and fermionic subspaces, the set of occupation numbers {n�k } uniquely identify a state.
We can, however, remove this restriction from eq.(VII.11), if we divide by the resulting
Q
over-counting factor (for bosons) of N !/( �k n�k !), i.e.,
Q
′
X
X � n� !
k k
=
.
N!
{�
k}
{�
k}
(Note that for fermions, the (−1)P factors cancel out the contributions from cases where
any n�k is larger than one.) Therefore,
Q
X � n� !
1
′
k k
Q
·
·
�{~x }|ρ|{~x}� =
N!
N ! �k n�k !
{�
k}
(VII.12)
!
N
X ηP ηP ′
X
¯ 2 kα2
h
exp −β
�{~x ′ }|P ′ {~k}��P {~k}|{~x}�.
2
Z
m
N
′
α=1
P,P
In the limit of large volume, the sums over {~k} can be replaced by integrals, and using the
plane wave representation of wavefunctions, we have
Z Y
N
X
V d3~kα
βh̄2 kα2
1
P P′
′
η η
exp −
�{~x }|ρ|{~x}� =
3
(2π)
2m
ZN (N !)2
′
α=1
P,P
h P
i

(VII.13)
~kP α · ~xα − ~kP ′ α · ~x ′ ) 
 exp −i N
(
α
α=1
×
.
N


V
We can order the sum in the exponent by focusing on a particular ~k-vector. Since
P
P
−1
β), where β = P α and α = P −1 β, we obtain
α f (P α)g(α) =
β f (β)g(P
"Z
#
N
3~
X
′
2 2
�
′ Y
d
k
1
xP −1 α −�
x ′−1
−βh̄ kα /2m
α −ikα · �
P
α
�{~x ′ }|ρ|{~x}� =
ηP ηP
e
.
2
3
(2π)
ZN (N !)
′
α=1
P,P
(VII.14)
141
The gaussian integrals in the square brackets are equal to
2 1
π ′
exp − 2 ~xP −1 α − ~x P ′−1 α
.
λ3
λ
Setting β = P −1 α in eq.(VII.14) gives
�{~x ′ }|ρ|{~x}� =
1
ZN
λ3N (N !)2
X
ηP ηP
P,P ′
′


N X
2
π
exp − 2
~xβ − ~x ′P ′−1 P β  .
λ
−1
Finally, we set Q = P ′−1 P , and use the results η P = η P , and η Q = η P
P
get (after performing P = N !)


N
X
X
2
1
π
η Q exp − 2
~xβ − ~x ′Qβ  .
�{~x ′ }|ρ|{~x}� =
3N
λ
ZN λ N !
Q
(VII.15)
β=1
′−1
P
′
= η P η P , to
(VII.16)
β=1
The canonical partition function, ZN , is obtained from the normalization condition
tr(ρ) = 1,
=⇒
as
ZN =
1
N !λ3N
Z Y
N
d3 ~xα
α=1
X
Q
Z Y
N
d3 ~xα �{~x}|ρ|{~x}� = 1,
α=1


N
X
π
2
η Q exp − 2
(~xβ − ~xQβ )  .
λ
(VII.17)
β=1
The quantum partition function thus involves a sum over N ! possible permutations. The
N
classical result ZN = V /λ3 /N !, is obtained from the term corresponding to no particle
exchange, Q ≡ 1. The division by N ! finally justifies the factor that was (somewhat
artificially) introduces in classical statistical mechanics to deal with the phase space of
identical particles. However, this classical result is only valid at very high temperature
and is modified by the quantum corrections coming from the remaining permutations.
As any permutation involves a product of factors exp[−π(~x1 − ~x2 )2 /λ2 ], its contributions
vanishes as λ → 0 for T → ∞.
The lowest order correction comes from the simplest permutation which is the ex­
change of two particles. The exchange of particles 1 and 2 is accompanied by a factor of
η exp[−2π(~x1 − ~x2 )2 /λ2 ]. As each of the possible N (N − 1)/2 pairwise exchanges gives the
same contribution to ZN , we get
ZN
1
=
N !λ3N
Z Y
N
α=1
3
d ~xα
N (N − 1)
2π
2
1+
η exp − 2 (~x1 − ~x2 ) + · · · .
2
λ
142
(VII.18)
For any α ≥ 2,
R
d3 ~xα = V ; in the remaining two integrations we can use the relative,
~r12 = ~x2 − ~x1 , and center of mass coordinates to get
ZN
Z
1
N (N − 1)
N
3
−2π�
r 212 /λ2
=
V
1+
+···
η d ~r12 e
N !λ3N
2V


!3
r
N
2πλ2
1
V
1 + N (N − 1) ·
=
η + · · · .
2V
4π
N ! λ3
(VII.19)
From the corresponding free energy,
F = −kB T ln ZN
e V
kB T N 2 λ3
= −N kB T ln 3 ·
−
· 3/2 η + · · · ,
λ N
2V
2
(VII.20)
the gas pressure is computed as
∂F N kB T
N 2 kB T λ3
ηλ3
P =−
· 5/2 η + · · · = nkB T 1 − 5/2 n + · · · .
=
−
∂V T
V
V2
2
2
(VII.21)
Note that the first quantum correction is equivalent to a second virial coefficient of
B2 = −
ηλ3
.
25/2
(VII.22)
The resulting correction to pressure is negative for bosons, and positive for fermions. In the
classical formulation, a second virial coefficient was obtained from a two-body interaction.
The classical potential V(~r ) that leads to the second virial coefficient in eq.(VII.22) is
obtained from
f (~r ) = e−β V (�r ) − 1 = ηe−2π�r /λ , =⇒
i
h
2
2
−2π�
r 2 /λ2
V(~r ) = −kB T ln 1 + ηe
≈ −kB T η e−2π�r /λ .
2
2
(VII.23)
(The final approximation corresponds to high temperatures, where only the first correc­
tion is important). Thus the effects of quantum statistics at high temperatures are ap­
proximately equivalent to introducing an interaction between particles. The interaction is
attractive for bosons, repulsive for fermions, and operates over distances of the order of
the thermal wavelength λ.
143
VII.C Grand Canonical Formulation
Calculating the partition function by performing all the sums in eq.(VII.17) is a
formidable task. Alternatively, we can compute ZN in the energy basis as


"
#
′
N
′
X
X
X
X
exp −β
E(~k)n(~k) .
exp −β
E(~kα ) =
ZN = tr e−β H =
α=1
{�
kα }
(VII.24)
�
k
{n~k }
These sums are still difficult to perform due to the restrictions of symmetry on the allowed
P
values of ~k or {n� }: The occupation numbers {n� } are restricted to � n� = N , and
k
k
k
k
n�k = 0, 1, 2, · · · for bosons, while n�k = 0 or 1 for fermions. As usual, the first constraint
can be removed by looking at the grand partition function,


∞
′
X
X
X
Qη (T, µ) =
eβµN
exp −β
E(~k)n�k 
N=0
{n~k }
η
X
Y
=
{n~k } �
k
�
k
(VII.25)
h
i
~
exp −β E(k) − µ n�k .
The sums over {n�k } can now be performed independently for each ~k, subject to the re­
strictions on occupation numbers imposed by particle symmetry.
• For fermions, n�k = 0 or 1, and
Q− =
Yh
i
1 + exp βµ − βE(~k) .
(VII.26)
�
k
• For bosons, n�k = 0, 1, 2, · · ·, and summing the geometric series gives
Q+ =
Yh
�
k
1 − exp βµ − βE(~k)
i−1
.
The results for both cases can be presented simultaneously as
X h
i
~
ln Qη = −η
ln 1 − η exp βµ − βE(k) ,
(VII.27)
(VII.28)
�
k
with η = −1 for fermions, and η = +1 for bosons.
In the grand canonical formulation, different one-particle states are occupied indepen­
dently, with a joint probability
n
o
h
i
1 Y
n(~k)
=
pη
exp −β E(~k) − µ n�k .
Qη
�
k
144
(VII.29)
The average occupation number of a state of energy E(~k) is given by
�n�k �η = −
∂ ln Qη
1
,
=
−1
β
E
~
z e (�k) − η
∂ βE(k)
(VII.30)
where z = exp(βµ). The average values of the particle number and internal energy are
then given by

X
X
1

�
=
=
�n
N

η
η
�
k


z −1 eβ E (�k) − η

�
�
k
k
.
X
X
~k)

E(

~
 Eη =
E(k)�n�k �η =


z −1 eβ E (�k) − η
�
k
(VII.31)
�
k
VII.D Non-relativistic Gas
Quantum particles are further characterized by a spin s. In the absence of a magnetic
field different spin states have the same energy, and a spin degeneracy factor, g = 2s + 1,
multiplies eqs.(VII.28)–(VII.31). In particular, for a non-relativistic gas in three dimen­
R
P
sions (E(~k) = h̄2 k 2 /2m, and � → V d3~k/(2π)3 ) these equations reduce to
k

Z
3~
2 2 k
k
d
βh̄
ln
Q
η


= ηg
ln 1 − ηz exp −
,
 βPη =

3

V
2m
(2π )



Z


Nη
d3~k
1

2 2
,
=g
nη ≡
3
V
(2π) z −1 exp βh̄ k − η

2m



Z


d3~k h
¯ 2 k2
1
Eη



=
g
ε
≡
.
η


V
(2π)3 2m z −1 exp βh̄2 k2 − η
(VII.32)
2m
To simplify these equations, we change variables to x = βh̄2 k 2 /(2m), so that
√
π 1/2 −1/2
2mkB T 1/2 2π 1/2 1/2
=
x ,
=⇒
dk =
x
dx.
k=
x
¯
h
λ
λ
Substituting into eqs.(VII.32) gives

Z
g 4π 3/2 ∞

1/2
−x

=
dx
x
1
−
ηze
−
η
ln
βP

η

2π 2 λ3

0


Z ∞


g
4
dx x3/2



√
=
, (integration by parts)

λ3 3 π 0 z −1 ex − η
Z ∞

dx x1/2
g 2


√
=
,
n


 η λ3 π 0 z −1 ex − η


Z ∞



g 2
dx x3/2

 βεη =
√
.
λ3 π 0 z −1 ex − η
145
(VII.33)
We now define two sets of functions by
η
fm
(z)
1
=
(m − 1)!
Z
∞
0
dx xm−1
.
z −1 ex − η
(VII.34)
For non-integer arguments, the function m! ≡ Γ(m + 1) is defined by the integral
R∞
√
dx xm e−x . In particular, from this definition it follows that (1/2)! = π/2, and
0
√
(3/2)! = (3/2) π/2. Eqs.(VII.33) now take the simple form

g η

βPη = 3 f5/2
(z),


λ


g η
(z),
nη = 3 f3/2
λ




 ε = 3P .
η
η
2
(VII.35)
These results completely describe the thermodynamics of ideal quantum gases as a function
of z. To find the equation of state Pη (nη , T ), we need to solve for z in terms of density.
η
(z).
This requires knowledge of the behavior of the functions fm
The high temperature, low density (non-degenerate) limit will be examined first. In
this limit, z is small, and
η
fm
(z)
1
=
(m − 1)!
=
=
=
1
(m − 1)!
∞
X
α=1
∞
X
α=1
Z ∞
−1
dx xm−1
1
=
dx xm−1 ze−x 1 − ηze−x
−1
x
z e −η
(m − 1)! 0
0
Z ∞
∞
X
α α+1
η
ze−x
dx xm−1
Z
0
η α+1 z α
η α+1
∞
1
(m − 1)!
Z
α=1
∞
dx xm−1 e−αx
0
z2
z3
z 4
zα
=
z
+
η
+
+
η
+ · · · .
2m
3m
4m
αm
(VII.36)
η
(z), and hence nη (z) and Pη (z), are indeed small
We thus find (self-consistently) that fm
as z → 0. Eqs.(VII.35) in this limit give,

nη λ3
z2
z3
z4
η


= f3/2 (z) = z + η 3/2 + 3/2 + η 3/2 + · · · ,

g
2
3
4
3
4
3
2


 βPη λ = f η (z) = z + η z + z + η z + · · · .
5/2
g
25/2 35/2
45/2
146
(VII.37)
The first of the above equations can be solved perturbatively, by the recursive procedure
of substituting the solution up to a lower order, as
z2
z 3
nη λ3
− η 3/2 − 3/2 − · · ·
g
2
3
2
η
nη λ3
nη λ3
−···
=
− 3/2
g
g
2
3
2 η
nη λ3
1
1
nη λ3
nη λ3
−
− ···.
− 3/2
=
+
g
4 33/2
g
g
2
z =
(VII.38)
Substituting this solution into the second leads to
2 3
nη λ3
η
nη λ3
1
1
nη λ3
− 3/2
+
−
g
g
4 33/2
g
2
2
3
3
η
1 nη λ3
1
nη λ3
nη λ3
+ ···.
+ 5/2
−
+ 5/2
g
8
g
g
2
3
βPη λ3
=
g
The pressure of the quantum gas can thus be obtained from the virial expansion,
"
Pη = n η k B T 1 −
η
25/2
nη λ3
g
+
1
2
− 5/2
8 3
nη λ3
g
2
#
+··· .
(VII.39)
The second virial coefficient B2 = −ηλ3 /(25/2g), agrees with eq.(VII.22) computed in
the canonical ensemble for g = 1. The natural (dimensionless) expansion parameter is
nη λ3 /g, and quantum mechanical effects become important when nη λ3 ≥ g; the quantum
degenerate limit. The behavior of fermi and bose gases is very different in this degenerate
limit of low temperatures and high densities, and the two cases will be discussed separately
in the following sections.
147
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8.333 Statistical Mechanics I: Statistical Mechanics of Particles
Fall 2013
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