Chapter 12

Quantum gases

In classical statistical mechanics, we evaluated thermodynamic relations often for an idealgas, which approximates a real gas in the highly diluted limit.

An important difference between classical and quantum mechanical many-body systemslies in the distinguishable character of their constituent particles, which gives rise to phe-nomena and concepts that are not present in classical physics. We start with a reminderof the basic notation.

12.1 Microcanonical ensemble

We consider an isolated system in thermodynamical equilibrium, with N particles andenergy between E and E +Δ.

Common eigenstates. An isolated ensemble in thermal equilibrium is characterized bya stationary distribution, i.e. by

[ρ, H] = 0 ,

which implies that the eigenstates of H

H |En� = En |En� , �En |Em� = δnm, �En| H |Em� = Emδnm

are also eigenstates of ρ:�En| ρ |Em� ∼ δnm .

Principle of uniform a priori probabilities.For an isolated system, the principle of “a priori” probability, as postulated in Sect. 7.5,states that all possible states of the system have the same probability. Therefore, we canwrite:

ρmicro =�

m

Pm |Em� �Em| Trρ = 1 , (12.1)

with

Pm =

�const. E < Em < E +Δ

0 otherwise

141

142 CHAPTER 12. QUANTUM GASES

where the constant is given by the normalization condition Trρ = 1.

Energy shell. The operator

PΔ =�

E<Em<E+Δ

|Em� �Em| , PΔ|Ψ� =�

E<Em<E+Δ

|Em��Em|� , (12.2)

projects to an arbitrary states to the energy shell. The trace Γ(E) of PΔ,

Γ(E) = Tr�PΔ

�=

�

E<Em<E+Δ

1 =number of states with energies

between E and E +Δ, (12.3)

is therefore the microcanonical phase-space volume.

Microcanonical density operator. The microcanonical density operator (12.1) is then

ρmicro =�

E<Em<E+Δ

Pm |Em� �Em| , Pm =1

Γ(E), (12.4)

with the expectation value of an operator B given in the microcanonical ensemble givenby

�B� =1

Γ(E)Tr

� �

E<Em<E+Δ

|Em� �Em| B�

Internal energy. The internal energy U is simply the expectation value of the energy,U = �H� = E, which one can show alternatively via

U ≡ 1

Γ(E)Tr

� �

E<Em<E+Δ

|Em��Em|H�

=1

Γ(E)

�

E<Em<E+Δ

Em ≈ E .

Entropy postulate. One possibility to ground quantum statistics is to postulated thatthe expression for the microcanonical entropy is within given by

S = kB lnΓ(E) . (12.5)

This definition is the quantum mechanical analogum of the postulate (8.5) of the entropywithin the classical microcanonical ensemble.

– There is no Gibbs paradox when the the definition (12.3) for the phase space volumeΓ(E) is used. Quantum mechanics takes the indistinguishability of identical particlescorrectly into account.

12.1. MICROCANONICAL ENSEMBLE 143

– Instead of postulating (12.5) one can use the information-theoretical expressionof the entropy and postulate, in accordance with Sect. 5.5.1, that the entropy ismaximal. This point of view will be touched upon Sect. 12.1.1.

Phase space volume. In analogy with classical statistics, we can define the phase spacevolume Φ(E) as

Φ(E) =�

Em≤E

1 ,

which measures the number of eigenstates of the Hamiltonian operator with energiessmall or equal to Em. The relation between the phase space density Γ(E), the phasespace volume Φ(E) and the density of states Ω(E) is

Γ(E) = Φ(E +Δ)− Φ(E), Ω(E) =δΦ

δE.

For small Δ0, the entropy (12.5) reduces to S = kB ln(Ω(E)Δ).

12.1.1 Maximum entropy principle

We may ground quantum statistic on information theoretical principles. One parts fromthe Shannon entropy I[P ],

I[P ] = −�

m

Pm lnPm , (12.6)

introduced in Sect. 5.5.3, demanding in a second step that the dimensionless entropy, S/kBcorresponds to a measure of the disorder present in the system and that this measure isquantified by I[P ].

Maximal entropy postulate. In Sect. 5.5.1 we have noted that the second law ofthermodynamics implies that maximum entropy principle, which states that irreversibleprocesses may only increase the entropy in isolated systems. Reversing the stance we mayuse the maximum energy principle as the basic postulate on which to ground quantumstatistics.

POSTULATE A statistical ensemble is thermally equilibratedwhen the Shannon entropy (12.6) is maximal.

This approach comes with the positive side effect that is holds for all three ensembles.

Maximal entropy distributions. In order to put the maximum entropy postulate atwork we need to know which are the maximal entropy distributions. For the case of themicrocanonical ensemble we need therefore to know which function p(y) maximizes

I[p] =

�p(y) ln

�p(y)

�dy, δI =

� �ln�p(y)

�+ 1

�� �� �

≡ 0

δp(y) dy , (12.7)

144 CHAPTER 12. QUANTUM GASES

where we have denoted the variation∗ of I[p] with δI. With the variation δp(y) of thedistribution function being arbitrary one has that the bracket on the right-hand-side ofδI in (12.7) needs to vanish for δI = 0. It follows that

p(y) = const. . (12.8)

Microcanonical entropy. The uniform distribution (12.8) reflects the principle of uni-form a priori probabilities entering the formulation of the microcanonical density matrix(12.4). for the entropy,

S = −kB�

E<Em<E+Delta

Pm ln(Pm), Pm =1

Γ(E),

it then follows

S = − kBΓ(E)

Γ(E) ln

�1

Γ(E)

�, S = kB lnΓ(E) , (12.9)

which reproduces the energy postulate (12.5). Γ(E) is here the volume of the energy shell.

12.1.2 Third law of thermodynamics

The third law of thermodynamics is of quantum mechanical nature:

“The entropy of a thermodynamical system at T = 0 is a universal constantthat can be chosen to be zero and this choice is independent of the values takenby the other state variables.”

Ground state degeneracy. For a system with a discrete energy spectrum, there is alowest energy state, the ground state. At T → 0, the system will go into this state.

If the groundstate is n-times degenerate, the entropy of the system at T = 0 is

S(T = 0) = kB lnn ,

n: degeneracy ≡ multiplicity.

For n = 1 (no degeneracy), S = 0.

Spontaneous symmetry breaking. The entropy S apparently doesn’t fulfill the thirdlaw of thermodynamics for a finite ground state degeneracy n > 1. However, there is noparadox.

∗ One uses variational calculus to derive Newton’s equations of motions,

d

dt

∂L

∂q=

∂L

∂q, W (L) =

�L (q, q) dt,

from the action W (L).

12.1. MICROCANONICAL ENSEMBLE 145

When n > 1, the groundstate is degenerate due to the existence of internal symmetriesof the Hamiltonian (for instance, spin rotational symmetry); at T = 0 the symmetry getsbroken through a phase transition that lets the entropy go to zero.

Spin rotational symmetry. The magnetization can point in an arbitrary directionfor a magnetic compound with spin rotational symmetry. In this case, one of the manypossible degenerate states is spontaneously selected.

12.1.3 Two-level system and the concept of negative tempera-ture

Consider an isolated system of N spin-1/2 particles, whichhave each two spin orientations, s = 1/2 and sz = ±1/2. Weassume that the two possible energy states ±� are occupiedby N± particles, with

N = N+ +N−, E = (N+ −N−) � ,

where E is the total energy. Obviously,

N+ =1

2

�N +

E

�

�, N− =

1

2

�N − E

�

�. (12.10)

Volume of the energy shell. The number of possible states with energy E and particlenumber N is given by the density of states Ω(E,N):

Γ(E,N) =

�N

N+

�=

N !

N+!N−!,

as there are�

NN+

�possibilities to select N+ particles out of a total of N indistinguishable

particles.

Microcanonical entropy. The entropy is then

S(E,N) = kb lnΓ(E,N) = kB ln

�N !

N+!N−!

�

= kB

�ln(N !)− ln(N+!)− ln(N−!)

�

≈ kB

�N lnN −N+ lnN+ −N− lnN−

�, (12.11)

where we have made use of the Stirling formula ln(N !) ≈ N lnN in the last step.

Temperature. The temperature of the collection of two-level is then

1

T=

∂S

∂E

����N

= −kB

�lnN+ + 1

�∂N+

∂E− kB

�lnN− + 1

�∂N−∂E

= −kB�lnN+ − lnN−

�/2� ,

146 CHAPTER 12. QUANTUM GASES

where we have used (12.11), namely that ∂N±/∂E = ±1/(2�). We then have

1

T=

kB2�

ln

�N−N+

�,

N+

N−= e−β2�, 2N± = N ± E/� . (12.12)

N− > N+ The normal state is characterized by E < 0, i.e. by a situation where thereare more particles in the lower energy level −� than in the higher energy level � > 0. Thetemperature is then positive, T > 0, as usual.

– All particles are in the lower energy level for N− = N , which corresponds to T = 0and β → ∞.

– The slope 1/T = ∂S/∂E diverges at E = −|�|.

N− = N+ For T → ∞, the particles distribute equally between the two levels: N+ =

N− = N/2. The total energy is then E = 0.

– The entropy (12.11) is then

S(E = 0, N) = kBN�lnN − ln(N/2)

�= kBN ln 2 .

The entropy per particle is given at T → ∞ by the logarithmof the number of degrees of freedom (in units of kB).

– The slope 1/T = ∂S/∂E vanishes at E = 0.

– The temperature T jumps from +∞ to −∞ as soon as more particles are in thehigher than the lower level.

N− < N+ Suppose that particles are excited to the higher energy level (for instance, by

light irradiation as in a Laser). If the particles remain in the excited state for a certain

12.2. CANONICAL ENSEMBLE 147

period of time (viz if the excited state is metastable), the situation N+ > N− is realized.This phenomenon is called inversion.

The definition (12.12) of the temperature,

1

T=

kB2�

ln

�N−N+

�,

becomes formally negative whenever N+ > N−.

– The concept of negative temperature makes only sense for systems with an upperbound for the energy.

– The inversion of the level occupation needs external pumping. Negative tempera-tures do therefore not occur for isolated systems. There is no true negative temper-ature.

– T → 0 from below when all the particles are in the higher level.

12.2 Canonical ensemble

In order to drive the density matrix ρ for the canonical ensemble one follow the routetaken in Sect. 9.1 for the classical ensemble. Here we will start, alternatively, from themaximum entropy postulate introduce in Sect. 12.1.1 as a possible grounding principlefor statistical mechanics.

Lagrange parameters. We start by determining the probability distribution p(y) max-imizing the Shannon information I[p] under the constraint of a given expectation value�y�. This is done within variational calculus by finding the maximum of

I[p]− λ�y� = −� �

ln�p(y)

�+ λ y

�p(y) dy , (12.13)

where λ is an appropriate Lagrange parameter.

Maximum entropy distribution. Variation of (12.13) yields

δ�I[p]− λ�y�

�= −

� �1 + ln

�p(y)

�+ λ y

�

� �� �≡ 0

δp(y) dy .

Compare the equivalent expression (12.7) without constraints. We therefore find that

p(y) ∼ e−λy, p(y) =e−λy

Z, Z =

�dy e−λy , (12.14)

with the normalization factor Z.

148 CHAPTER 12. QUANTUM GASES

System in thermal contact with a heat bath.The energy fluctuates for a system A1 in thermal equi-librium with a reservoir A2, which is the setup of thecanonical ensemble. We therefore need to find the dis-tribution function Pm maximizing

S − λ�E� = −kB�

m

Pm ln(Pm)− kBλ�

m

EmPm ,

where we rescaled the Lagrange parameter λ by kB. Our result (12.14) then tells us, that

Pm =e−λEm

Z, Z =

�

m

e−λEm . (12.15)

What remains to be done is to determine the physical significance of the λ.

Temperature as a Lagrange parameter. We will now show that

λ =1

kBT, U =

�m Eme

−βEm

Z, Z =

�

m

e−βEm , (12.16)

which implies that the (inverse) temperature β can be defined as the Lagrange parameterneeded to regulate the magnitude of the internal energy U = �E�.Free energy. The relation (12.16) is verified by showing that there exist for λ = β afunction F fulfilling with

− ∂F

∂T= S ≡ −kB

�

m

Pm ln(Pm), F = U − TS . (12.17)

the defining thermodynamic relations of the free energy. We use ln(Pm) = −λEm − lnZ,which allows to rewrite the entropy as

TS = kBT�

Pm

�βEm + lnZ

�= λβ����

=1

�E�����=U

+ kBT lnZ� �� �≡−F

. (12.18)

Defining the free energy as F = −kBT lnZ is therefore consistent with F = U − TS.The first condition of (12.17) is verified by evaluating

−∂F

∂T=

∂

∂T

�kBT lnZ

�= kB lnZ +

kBT

Z

��

m

e−βEm(−Em)

�−1

kBT 2

= kB lnZ +1

T�E�

≡ S ,

where we compared with (12.18) in the last step.

12.2. CANONICAL ENSEMBLE 149

Specific heat. With (4.8) and (12.18), namely

�∂S

∂T

�

V,N

=CV

T, S = kB lnZ +

1

T

�Em

e−βEm

Z

we obtain�∂S

∂T

�

V,N

=kBZ

�

m

(−Em)e−βEm

� −1

kBT 2

�+

�−1

T 2

��Em

e−βEm

Z

+1

T

�(−E2

m)e−βEm

Z

� −1

kBT 2

�+

�E�T

�

m

(−Em)e−βEm

−Z

� −1

kBT 2

�,

which leads to

CV =1

kbT 2

��E2

�−

�E�2�

. (12.19)

The quantummechanical result (12.19) for the specific heat CV is is hence identical to theclassical expression (9.23).

Canonical density matrix. Using that the eigenstates |Em� of Hamilton operator Hform a complete basis,

H|Em� = Em|Em�,�

m

|Em� �Em| = 1 ,

we write the canonical density matrix operator ρ as

ρ =1

Z

�

m

e−βEm |Em� �Em| =e−βH

Z

�

m

|Em� �Em|� �� �

1

,

that is as

ρ =e−βH

Z, Z = Tr

�e−βH

�, (12.20)

where we have used that the canonical partition function as Z = Z(T, V,N) =�

m e−βEm .

12.2.1 Quantum mechanical harmonic oscillators

As an application we consider N localized independent linear harmonic oscillators withfrequency ω in thermal equilibrium at temperature T . Such a system would describe, forinstance, the radiation energy of a black body.

Spectrum. A single oscillator is described by

HΨn = εnΨn, εn = hω

�n+

1

2

�,

where n is the occupation number. The spectrum �n is equally spaced.

150 CHAPTER 12. QUANTUM GASES

Partition function for one oscillator. Then, the partition function in the canonicalensemble for one oscillator is given by

Z(T, V,N = 1) = Tre−βH =�

n

e−βεn =∞�

n=0

e−βhω(n+ 12)

= e−βhω2

∞�

n=0

�e−βhω

�n,

which is a geometrical series of the form

∞�

n=0

rn =1

1− r.

Therefore,

Z(T, V, 1) =e−

βhω2

1− e−βhω=

1

eβhω2 − e−

βhω2

=1

2 sinh�βhω2

� .

Partition function for N oscillators. For N particles it then follows that

Z(T, V,N) =�Z(T, V, 1)

�N=

�2 sinh

�βhω

2

��−N

.

The above expression is valid only because we assumed that the oscillators don’t interactwith each other and that all eigenfrequencies are idential.

Free energy. The free energy F = −kBT ln(Z) is

F (T, V,N) = −NkBT ln

�e−βhω/2

1− e−βhω

�

=N

2hω

� �� �zero-point energyof N oscillators

+NkBT ln�1− e−βhω

�.

For the derivatives we obtain

µ =

�∂F

∂N

�

T,V

=F

N, P = −

�∂F

∂V

�

T,N

= 0 .

Note that the pressure P vanishes due to the fact that the oscillators are localized in thesense that the eigenfrequencies ω do not depend on the volume V .

Entropy. The entropy is

S = −�∂F

∂T

�

V,N

= −NkB ln�1− e−βhω

�−NkBT

hωe−βhω

1− e−βhω

� −1

kBT 2

�

= NkB

�βhω

eβhω − 1− ln

�1− e−βhω

��.

12.3. GRAND CANONICAL ENSEMBLE 151

Internal energy. The internal energy U = F + TS is

U = Nhω

�1

2+

1

eβhω − 1

�, (12.21)

which can be written as

U = Nhω

�1

2+ �n�

�, �n� =

1

eβhω − 1,

where �n� is the particle occupation number. We will see thus expression again when weintroduce the Bose gas.

– The T → 0 limit,

U → N

2hω ,

gives the ground-state energy of a system of N quantum oscillators (vacuum fluc-tuations).

– For T → ∞ one recovers withU → NkBT

the expression for the energy N classical oscillators.

12.3 Grand canonical ensemble

A grand canonical ensemble may exchangeboth energy and particles with a reser-voir. The eigenstates depend therefor onthe number N of particles present:

H|Em(N)� = Em(N)|Em(N)�,N |Em(N)� = N |Em(N)�,

1 =�

N,m

|Em(N)��Em(N)| ,

where N is the particle number operator.

Maximum entropy postulate. In order to apply the maximum entropy principle tothe grand canonical ensemble we have therefore to find the distribution function Pm(N)maximizing the Shannon entropy under the constraints that the expectation values of theenergy and of the particle number,

E = �H�, N = �N� ,are given. Following the calculation performed in Sect. 12.2 for the canonical ensemblewe look for the distribution Pm(N) maximizing

kB�I[P ]− λE�H� − λN�N�

�= −kB

�

N,m

Pm(N)�lnPm(N) + λEEm(N) + λNN

�,

152 CHAPTER 12. QUANTUM GASES

where I[P ] is the Shannon entropy. The solution is, in analogy to (12.15),

Pm(N) =e−λEEm−λNN

Z , Z =�

N,m

e−λEEm−λNN . (12.22)

The chemical potential as an Lagrange parameter. The physical significance ofthe two Lagrange parameters involved are given by

λE ≡ β, λN ≡ −βµ . (12.23)

The first equivalence was shown in Sect. 12.2. The second term in (12.23) implies that thechemical potential µ may be defined, modulo a factor (−β), a the Lagrange parameterregulating the particle number.

Density matrix. Using (12.23) we find

ρ =e−β(H−µN)

Z , Z = Tr�e−β(H−µN)

�(12.24)

for the grand canonical density matrix ρ and

Ω(T, V, µ) = −kBT lnZ(T, V, µ) (12.25)

respectively for the grand canonical potential Ω.

Particle number. In order to verify (12.23) we confirm that

N = −�∂Ω

∂µ

�

T,V

= kBT∂

∂µln

��

N,m

e−β(Em(N)−µN)

�

� �� �β(

�N,m Ne−β(Em(N)−µN))/Z =β�N�

holds.

12.3.1 Entropy and the density matrix operator

We showed in Sect. 12.1.1 that the entropy S can be written within the microcanonicalensemble in terms of the density matrix operator ρ as

S = −kBTr(ρ ln ρ) = −kB�ln ρ� , (12.26)

i.e., the entropy is given by the expectation value of the negative logarithm of the densitymatrix operator ρ.

– Note that since the eigenvalues of ρ are probabilities, i.e., they are between 0 and1, the logarithm in Eq. (12.26) will therefore be negative and S positive.

12.4. PARTITION FUNCTION OF IDEAL QUANTUM GASES 153

– For the canonical ensemble we used (12.2) as the starting point for the maximumenergy principle and showed that the resulting free energy F = F (T, V,N) is con-sistent with the requirement that F = U − TS and S = −∂F/∂T .

An analogous consistency check as done for the canonical ensemble can be performed alsofor the grand canonical ensemble.

Consistency check. For the grand canonical ensemble we have Ω(T, V, µ) = −kBT lnZand hence

ρ =1

Z e−β(H−µN) = eβΩe−β(H−µN) Z = e−βΩ .

With F = U − TS and Ω = F − µN we then have

−kB�ln ρ� = − 1

T

�Ω− �H�+ µ�N�

�

= − 1

T(F − U) = S .

Q.e.d.

12.4 Partition function of ideal quantum gases

The adequate ensemble to deal with quantum gases is the grand canonical ensemble:

Z(T, V, µ) = Tr e−β(H−µN) . (12.27)

Particle number operator. The particle number operator N entering (12.27) is givenby

N =�

k

c†kck, �c†kck� ∈ [0, 1]� �� �fermions

, �a†kak� = 0, 1, 2, . . .� �� �bosons

(12.28)

where c†kck is the operator measuring the occupancy of a plane wave with momentum hk.Two fermions may never occupy the same orbital.

Hamiltonian. The Hamilton operator H is

H =�

k

�k c†kck, �k =

(hk)2

2m, k = |k| (12.29)

for a gas of a non-interacting gas. �k is the dispersion relation.

Additive Hamilton operators. The particle number operator N is always additive.The exponential in the partition function (12.27) factorizes,

e−β(H−µN) =�

k

e−β(�k−µ)c†kck (12.30)

when the Hamilton operators is also additive. This holds because particle number oper-ators for different orbitals commute for both fermions and bosons,

�c†k1

ck1, c†k2

ck2

�= 0,

�a†k1

ak1, a†k2

ak2

�= 0 ,

154 CHAPTER 12. QUANTUM GASES

where k1 �= k2.

Partition function. We define

nk =

��c†kck� fermions

�a†kak� bosons(12.31)

and find with (12.30) that

Z(T, V, µ) =

�

nk1

e−βnk1(�k1−µ)

�

nk2

e−βnk2(�k2−µ)

· · ·

=�

k

��

nk

e−βnk(�k−µ)

�(12.32)

holds for the partition function Z(T, V, µ) of non-interaction particles. Note that thepartition function factorizes only for the grand canonical partition function.

Bosons. The occupation numbers nr runs over all non-negative integers for a system ofbosons, so that the parenthesis of (12.32) reduces to a geometrical series†:

Z(T, V, µ) =�

r

�1

1− e−β(�r−µ)

�

bosons

(12.33)

We have used here an index r for the sum over orbitals characterized by an energy �r.One may identify r with the wavevector k whenever needed. Note that (12.33) impliesthat �r − µ > 0. The chemical potential µ must hence be small than any eigenenergy �r.

Fermions. The orbital occupations number nr takes only the values nr = 0 and nr = 1for a system of fermions. This implies that

Z(T, V, µ) =�

r

�1 + e−β(�r−µ)

�

fermions

(12.34)

Once Z has been determined one can obtain the remaining thermodynamic properties ofideal quantum gases.

12.5 Thermodynamic properties of ideal quantum gases

Using the index (+) and (–) for bosons and fermions respectively we find with (12.33)

Ω(+)(T, V, µ) = kBT�

r

ln�1− e−β(�r−µ)

�

bosons

(12.35)

† The sum of a geometrical series is�∞

n=0 xn = 1/(1− x).

12.5. THERMODYNAMIC PROPERTIES OF IDEAL QUANTUM GASES 155

for the bosonic grand canonical potential Ω(+) = −kBT lnZ(+) and

Ω(−)(T, V, µ) = −kBT�

r

ln�1 + e−β(�r−µ)

�

fermions

(12.36)

for the case of fermions. The volume dependence results from the sum�

r over orbitals.

Bose-Einstein distribution. The expectation value of the particle number operator Nis

�N�(+) = − ∂

∂µΩ(+)(T, V, µ) =

�

r

1

eβ(�r−µ) − 1,

which allows to define the Bose-Einstein distribution

�nr�(+) =1

eβ(�r−µ) − 1Bose-Einstein

(12.37)

where nr = a†kak denotes the number operator for the orbital r.

Fermi-Dirac distribution. For Fermions one finds correspondingly

�N�(−) = − ∂

∂µlnΩ(−)(T, V, µ) =

�

r

1

eβ(�r−µ) + 1.

The expectation value of the orbital occupation,

�nr�(−) =1

eβ(�r−µ) + 1Fermi-Dirac

(12.38)

is the Fermi-Dirac distribution function.

Thermal equation of state. The Thermal equation of state is

PV = −Ω, PV = kBT lnZ(±)(T, V, µ) .

according to Sect. 5.4, which holds also for ideal quantum gases.

Internal energy. The internal energy U = U (±) is given with (5.21) by

U − µN =∂

∂β

�βΩ

�, U (±) =

�

r

�r �nr�(±) ,

where the orbital occupation number �nr�(±) is given by (12.37) and (12.38) for bosonsand respectively for fermions. The internal energy reduces hence to the sum over themean orbital energies, For the derivation of this intuitive results we consider the case ofbosons:

U (+) − µ�N�(+) = − ∂

∂βlnZ(+)(T, V, µ) =

�

r

�r − µ

eβ(�r−µ) − 1=

�

r

n(+)r (�r − µ) .

156 CHAPTER 12. QUANTUM GASES

12.5.1 Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac dis-tribution

The three distributions

nMB(�r) = e−β(�r−µ), nBE(�r) =1

eβ(�r−µ) − 1, nFD(�r) =

1

eβ(�r−µ) + 1

quantify the occupation of an energy level for a non-interacting gas.

nMB : Maxwell-Boltzmann (classical grand canonical ensemble)

nBE : Bose-Einstein (quantum)

nFD : Fermi-Dirac (quantum)

For large one-particle energies, i.e., for �r − µ >� kBT , both the Bose-Einstein andthe Fermi-Dirac distribution functions converge to the Maxwell-Boltzmann distributionfunction.

All three distribution functions are function only of x = β(�r − µ).

Chemical potential – bosons. For bosons the lowest energy state �0 sets an upperlimit for the chemical potential. The occupation

nBE0 (�0) =

1

eβ(�0−µ) − 1, �0 ≤ �r

of the lowest energy state would be otherwise negative, i.e. if µ would be larger than �0.

Bose-Einstein condensation. There are cases when the chemical potential µ ap-proaches the lowest energy level �0 fro below. The occupation of the lowest energy levelthen diverges:

limµ→�0

nBE0 (�0) = lim

µ→�0

1

eβ(�0−µ) − 1→ ∞ . (12.39)

This phenomenon is called Bose-Einstein condensation.

– A finite fraction αN , with 0 ≤ α ≤ 1, of all particles occupy the identical state,namely �0 when (12.39) occurs. The remaining fraction (1−α)N of particles occupythe higher level.

– 4He is a famous example of a real gas undergoing a Bose-Einstein condensation.The residual interactions between the Helium atoms are however substantial.

12.5. THERMODYNAMIC PROPERTIES OF IDEAL QUANTUM GASES 157

Chemical potential – fermions.The chemical potential regulates thenumber of particles,

0 ≤ nFD ≤ 1 (Pauli principle) .

µ can have any value. The T → 0limit of nFD is a step function:

limT→0

1

eβ(�r−µ) + 1=

�1 �r < µ0 �r > µ

�F = limT→0 µ(T ) is the Fermi energy.

12.5.2 Particle number fluctuations

We restrict ourselves to a single orbital with orbital energy �r and write the expectationvalue of the particle number as

�nr� =1

Z�

nr

nr e−nrxr , Z =

�

nr

e−nrxr , xr = β(�r − µ) .

We leave the allowed values of nr, which depend on the statistics, open. We have

�nr� = − ∂

∂xr

lnZ, �n2r� =

1

Z�

nr

n2r e

−nrxr ,

where the second relation defines the second moment �n2r� of the particle number operator

nr We evaluate

∂

∂xr

�nr� =∂

∂xr

1

Z�

nr

nr e−nrxr

=

�1

Z�

nr

nr e−nrxr

��1

Z�

nr

nr e−nrxr

�− 1

Z�

nr

n2r e

−nrxr .

Particle number fluctuations. Using above expressions we find that the relation

�n2r� − �nr�2 = − ∂

∂xr

�nr� (12.40)

holds independently of the statistics considered. The variance of the particle numberoperator is proportional to the (negative) derivative the its expectation value.

Relative particle number fluctuations. Carrying the derivation on the right-handside of (12.40) out for the our three statistics (MB, BE, FD), we obtain

�n2r� − �nr�2�nr�2

= eβ(�r−µ),�n2

r� − �nr�2�nr�2

=1

�nr�− c

158 CHAPTER 12. QUANTUM GASES

for the relative particle number fluctuations .

cMB = 0 for classical particles (Maxwell-Boltzmann)cBE = −1 for bosons (Bose-Einstein)cFD = +1 for fermions (Fermi-Dirac)

The particle fluctuations for bosons are larger than those for classical particles while forfermions they are smaller. This result reflect the intuition that The reason for that is thatthe Pauli principle hinders the change of a particle state for the case of fermions but notfor bosons.