Chapter 12

Quantum gases

In classical statistical mechanics, we evaluated thermodynamic relations often for an ideal gas, which approximates 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, which gives rise to phe- nomena and concepts that are not present in classical physics. We start with a reminder of the basic notation.

12.1 Microcanonical ensemble

We consider an isolated system in thermodynamical equilibrium, withN particles and energy betweenE andE+Δ. Common eigenstates. An isolated ensemble in thermal equilibrium is characterized by a stationary distribution, i.e. by [ˆρH,ˆ]=0, which implies that the eigenstates of Hˆ

Hˆ E =E E , E E =δ , E Hˆ E =E δ | n� n | n� � n | m� nm � n| | m� m nm are also eigenstates of ˆρ: E ˆρ E δ . � n| | m� ∼ nm 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 can write: ˆρ = P E E Trˆρ=1, (12.1) micro m | m� � m| m � with const. E

141 142 CHAPTER 12. QUANTUM GASES

where the constant is given by the normalization condition Trˆρ = 1. Energy shell. The operator

Pˆ = E E , Pˆ Ψ = E E Ψ , (12.2) Δ | m� � m| Δ| � | m�� m| � E

projects to an arbitrary states to the energy shell. The traceΓ(E) of PˆΔ,

number of states with energies Γ(E) = Tr Pˆ = 1 = , (12.3) Δ betweenE andE+Δ Evolume. Microcanonical density operator. The microcanonical density operator (12.1) is then

1 ˆρ = P E E ,P = , (12.4) micro m | m� � m| m Γ(E) E

with the expectation value of an operator Bˆ given in the microcanonical ensemble given by 1 Bˆ = Tr E E Bˆ � � Γ(E) | m� � m| �E

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

S=k B lnΓ(E) . (12.5)

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

– There is no Gibbs paradox when the the definition (12.3) for the phase space volume Γ(E) is used. Quantum mechanics takes the indistinguishability of identical particles correctly into account. 12.1. MICROCANONICAL ENSEMBLE 143

– Instead of postulating (12.5) one can use the information-theoretical expression of the entropy and postulate, in accordance with Sect. 5.5.1, that the entropy is maximal. 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 space volumeΦ(E) as Φ(E) = 1,

Em E �≤ which measures the number of eigenstates of the Hamiltonian operator with energies small or equal toE m. The relation between the phase space densityΓ(E), the phase space volumeΦ(E) and the density of statesΩ(E) is

δΦ Γ(E) =Φ(E+Δ) Φ(E),Ω(E)= . − δE

For smallΔ0, the entropy ( 12.5) reduces toS=k B ln(Ω(E)Δ).

12.1.1 Maximum entropy principle We may ground quantum statistic on information theoretical principles. One parts from the Shannon entropyI[P], I[P]= P lnP , (12.6) − m m m � introduced in Sect. 5.5.3, demanding in a second step that the dimensionless entropy, S/kB corresponds to a measure of the disorder present in the system and that this measure is quantified byI[P]. Maximal entropy postulate. In Sect. 5.5.1 we have noted that the second law of thermodynamics implies that maximum entropy principle, which states that irreversible processes may only increase the entropy in isolated systems. Reversing the stance we may use the maximum energy principle as the basic postulate on which to ground quantum statistics.

POSTULATE A statistical ensemble is thermally equilibrated when 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 at work we need to know which are the maximal entropy distributions. For the case of the microcanonical ensemble we need therefore to know which functionp(y) maximizes

I[p] = p(y) ln p(y) dy,δI= ln p(y) + 1 δp(y) dy , (12.7)

� � 0 � � � � ≡ � � � �� � 144 CHAPTER 12. QUANTUM GASES

where we have denoted the variation∗ ofI[p] withδI. With the variationδp(y) of the distribution function being arbitrary one has that the bracket on the right-hand-side of δI in (12.7) needs to vanish forδI = 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,

1 S= k P ln(P ),P = , − B m m m Γ(E) E

12.1.2 Third law of thermodynamics The third law of thermodynamics is of quantum mechanical nature:

“The entropy of a thermodynamical system atT=0 is a universal constant that can be chosen to be zero and this choice is independent of the values taken by the other state variables.”

Ground state degeneracy. For a system with a discrete energy spectrum, there is a lowest energy state, the ground state. AtT 0, the system will go into this state. → If the groundstate isn-times degenerate, the entropy of the system atT = 0 is

S(T = 0) =k B lnn,

n: degeneracy multiplicity. ≡ Forn = 1 (no degeneracy),S = 0. Spontaneous symmetry breaking. The entropyS apparently doesn’t fulfill the third law of thermodynamics for afinite ground state degeneracyn> 1. However, there is no paradox.

∗ One uses variational calculus to derive Newton’s equations of motions, d ∂L ∂L = ,W(L) = L(q,˙q) dt, dt ∂˙q ∂q � from the actionW(L). 12.1. MICROCANONICAL ENSEMBLE 145

Whenn> 1, the groundstate is degenerate due to the existence of internal symmetries of the Hamiltonian (for instance, spin rotational symmetry); atT = 0 the symmetry gets broken through a phase transition that lets the entropy go to zero. Spin rotational symmetry. The magnetization can point in an arbitrary direction for a magnetic compound with spin rotational symmetry. In this case, one of the many possible degenerate states is spontaneously selected.

12.1.3 Two-level system and the concept of negative tempera- ture Consider an isolated system ofN spin-1/2 particles, which have each two spin orientations,s=1/2 ands z = 1/2. We assume that the two possible energy states � are± occupied byN particles, with ± ±

N=N + +N ,E=(N + N )�, − − − whereE is the total energy. Obviously,

1 E 1 E N+ = N+ ,N = N . (12.10) 2 � − 2 − � � � � � Volume of the energy shell. The number of possible states with energyE and particle numberN is given by the density of statesΩ(E,N):

N N! Γ(E,N)= = , N+ N+!N ! � � − as there are N possibilities to selectN particles out of a total ofN indistinguishable N+ + particles. � � Microcanonical entropy. The entropy is then

N! S(E,N)=k b lnΓ(E,N) =k B ln N+!N ! � − � =k B ln(N!) ln(N +!) ln(N !) − − − � � k B 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 ∂S ∂N+ ∂N = = k B lnN + + 1 k B lnN + 1 − T ∂E − ∂E − − ∂E �N � � � � � � = k B lnN + lnN /2�, � − − − � � 146 CHAPTER 12. QUANTUM GASES

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

1 kB N N+ β2� = ln − , =e − ,2N =N E/�. (12.12) T 2� N+ N ± ± � � −

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

– All particles are in the lower energy level forN =N, which corresponds toT=0 − andβ . →∞ – The slope 1/T=∂S/∂E diverges atE= � . −| |

N =N + ForT , the particles distribute equally between the two levels:N + = − →∞ N =N/2. The total energy is thenE = 0. − – The entropy (12.11) is then

S(E=0,N)=k N lnN ln(N/2) =k N ln 2. B − B � � The entropy per particle is given atT by the logarithm →∞ of the number of degrees of freedom (in units ofk B).

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

– The temperatureT jumps from + to as soon as more particles are in the higher 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 situationN + > N is realized. − This phenomenon is called inversion.

The definition (12.12) of the temperature,

1 kB N = ln − , T 2� N � + �

becomes formally negative wheneverN + > N . −

– The concept of negative temperature makes only sense for systems with an upper bound 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 ˆρ for the canonical ensemble one follow the route taken in Sect. 9.1 for the classical ensemble. Here we will start, alternatively, from the maximum entropy postulate introduce in Sect. 12.1.1 as a possible grounding principle for statistical mechanics. Lagrange parameters. We start by determining the probability distributionp(y) max- imizing the Shannon informationI[p] under the constraint of a given expectation value y . This is done within variational calculus byfinding the maximum of � � I[p] λ y = ln p(y) +λy p(y) dy , (12.13) − � � − � � � � � whereλ is an appropriate Lagrange parameter. Maximum entropy distribution. Variation of (12.13) yields

δ I[p] λ y = 1 + ln p(y) +λy δp(y) dy . − � � − � � � � � 0 � � ≡ Compare the equivalent expression (12.7�) without�� constraints.� We thereforefind that

λy λy e− λy p(y) e − , p(y) = ,Z= dy e− , (12.14) ∼ Z � with the normalization factorZ. 148 CHAPTER 12. QUANTUM GASES

System in thermal contact with a heat bath. The energyfluctuates for a systemA 1 in thermal equi- librium with a reservoirA 2, which is the setup of the canonical ensemble. We therefore need tofind the dis- tribution functionP m maximizing

S λ E = k P ln(P ) k λ E P , − � � − B m m − B m m m m � �

where we rescaled the Lagrange parameterλ byk B. Our result (12.14) then tells us, that

λEm e− λEm P = ,Z= e− . (12.15) m Z m � What remains to be done is to determine the physical significance of theλ. Temperature as a Lagrange parameter. We will now show that

βEm 1 m Eme− βEm λ= ,U= ,Z= e− , (12.16) k T Z B m � � which implies that the (inverse) temperatureβ can be defined as the Lagrange parameter needed to regulate the magnitude of the internal energyU= E . � � Free energy. The relation (12.16) is verified by showing that there exist forλ=βa functionF fulfilling with

∂F =S k P ln(P ),F=U TS. (12.17) − ∂T ≡− B m m − m �

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

TS=k T P βE + lnZ = λβ E +k T lnZ . (12.18) B m m � � B = 1 =U F � � � ≡ −

Defining the free energy asF= k BT lnZ is therefore���� ���� consistent� �� with� F=U TS. Thefirst condition of ( 12.17) is− verified by evaluating −

∂F ∂ kBT βEm 1 = k T lnZ =k lnZ+ e− ( E ) − − ∂T ∂T B B Z − m k T 2 � m � B � � 1 � =k lnZ+ E B T � � 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

βEm ∂S CV 1 e− = ,S=k B lnZ+ Em ∂T V,N T T Z � � � we obtain

βEm ∂S kB βEm 1 1 e− = ( E )e− − + − E ∂T Z − m k T 2 T 2 m Z V,N m B � � � � � � � � βEm βEm 1 e− 1 E e− 1 + ( E2 ) − + � � ( E ) − , T − m Z k T 2 T − m Z k T 2 B m B � � � � − � � which leads to 1 2 2 CV = 2 E E . (12.19) kbT − �� � � � � The quantummechanical result (12.19) for the specific heatC V is is hence identical to the classical expression (9.23).

Canonical density matrix. Using that the eigenstates E m of Hamilton operator Hˆ form a complete basis, | �

Hˆ E =E E , E E = 1ˆ, | m� m| m� | m� � m| m � we write the canonical density matrix operator ˆρ as

βHˆ 1 βEm e− ˆρ= e− E E = E E , Z | m� � m| Z | m� � m| m m � � 1ˆ that is as � �� � βHˆ e− βHˆ ˆρ= ,Z = Tr e− , (12.20) Z � � βEm where we have used that the canonical partition function asZ=Z(T,V,N)= m e− . � 12.2.1 Quantum mechanical harmonic oscillators As an application we considerN localized independent linear harmonic oscillators with frequencyω in thermal equilibrium at temperatureT . Such a system would describe, for instance, the radiation energy of a black body. Spectrum. A single oscillator is described by 1 HˆΨ =ε Ψ ,ε = ¯hω n+ , n n n n 2 � �

wheren 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 canonical ensemble for one oscillator is given by

ˆ ∞ 1 βH βεn β¯hω(n+ ) Z(T,V,N = 1) = Tre− = e− = e− 2 n n=0 � � ∞ β¯hω β¯hω n =e − 2 e− , n=0 � � � which is a geometrical series of the form

∞ 1 rn = . 1 r n=0 � − Therefore, β¯hω e− 2 1 1 Z(T,V, 1) = β¯hω = β¯hω β¯hω = β¯hω . 1 e − e 2 e − 2 2 sinh − − 2 Partition function forN oscillators. ForN particles it then follows� � that

N N β¯hω − Z(T,V,N)= Z(T,V, 1) = 2 sinh . 2 � � �� � � The above expression is valid only because we assumed that the oscillators don’t interact with each other and that all eigenfrequencies are idential. Free energy. The free energyF= k T ln(Z) is − B e β¯hω/2 F(T,V,N) = Nk T ln − − B 1 e β¯hω � − − � N β¯hω = ¯hω +Nk T ln 1 e − . 2 B − zero-point energy � � ofN oscillators � �� � For the derivatives we obtain ∂F F ∂F µ= = ,P= = 0. ∂N N − ∂V � �T,V � �T,N Note that the pressureP vanishes due to the fact that the oscillators are localized in the sense that the eigenfrequenciesω do not depend on the volumeV. Entropy. The entropy is

β¯hω ∂F β¯hω ¯hω−e 1 S= = Nk ln 1 e − Nk T − − ∂T − B − − B 1 e β¯hω k T 2 � �V,N − − � B � � � β¯hω β¯hω = Nk ln 1 e − . B eβ¯hω 1 − − � − � � � 12.3. GRAND CANONICAL ENSEMBLE 151

Internal energy. The internal energyU=F+TS is 1 1 U=N¯hω + , (12.21) 2 eβ¯hω 1 � − � which can be written as 1 1 U=N¯hω + n , n = , 2 � � � � eβ¯hω 1 � � − where n is the particle occupation number. We will see thus expression again when we introduce� � the Bose gas. – TheT 0 limit, → N U ¯hω, → 2 gives the ground-state energy of a system ofN quantum oscillators (vacuumfluc- tuations). – ForT one recovers with →∞ U Nk T → B the expression for the energyN classical oscillators.

12.3 Grand canonical ensemble

A grand canonical ensemble may exchange both energy and particles with a reser- voir. The eigenstates depend therefor on the numberN of particles present: Hˆ E (N) =E (N) E (N) , | m � m | m � Nˆ E (N) =N E (N) , | m � | m � 1ˆ= E (N) E (N) , | m �� m | N,m � where Nˆ is the particle number operator. Maximum entropy postulate. In order to apply the maximum entropy principle to the grand canonical ensemble we have therefore tofind the distribution functionP m(N) maximizing the Shannon entropy under the constraints that the expectation values of the energy and of the particle number, E= Hˆ ,N= Nˆ , � � � � are given. Following the calculation performed in Sect. 12.2 for the canonical ensemble we look for the distributionP m(N) maximizing

k I[P] λ Hˆ λ Nˆ = k P (N) lnP (N)+λ E (N)+λ N , B − E� �− N � � − B m m E m N N,m � � � � � 152 CHAPTER 12. QUANTUM GASES

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

λE Em λN N e− − λ Em λ N P (N)= , = e− E − N . (12.22) m Z N,m Z � The chemical potential as an Lagrange parameter. The physical significance of the two Lagrange parameters involved are given by

λ β,λ βµ. (12.23) E ≡ N ≡ − Thefirst equivalence was shown in Sect. 12.2. The second term in (12.23) implies that the chemical potentialµ may be defined, modulo a factor ( β), a the Lagrange parameter regulating the particle number. − Density matrix. Using (12.23) wefind

β(Hˆ µNˆ) e− − β(Hˆ µNˆ) ˆρ= , = Tr e− − (12.24) Z Z � � for the grand canonical density matrix ˆρ and

Ω(T, V, µ) = k T ln (T, V, µ) (12.25) − B Z respectively for the grand canonical potentialΩ. Particle number. In order to verify (12.23) we confirm that

∂Ω ∂ β(Em(N) µN) N= =k T ln e− − − ∂µ B ∂µ T,V �N,m � � � � β Ne β(Em(N) µN) / =β Nˆ ( N,m − − ) Z � � � � �� � holds.

12.3.1 Entropy and the density matrix operator We showed in Sect. 12.1.1 that the entropyS can be written within the microcanonical ensemble in terms of the density matrix operator ˆρ as

S= k Tr(ˆρ ln ˆρ) = k ln ˆρ , (12.26) − B −B� � i.e., the entropy is given by the expectation value of the negative logarithm of the density matrix operator ˆρ.

– Note that since the eigenvalues of ˆρ are probabilities, i.e., they are between 0 and 1, the logarithm in Eq. (12.26) will therefore be negative andS positive. 12.4. PARTITION FUNCTION OF IDEAL QUANTUM GASES 153

– For the canonical ensemble we used (12.2) as the starting point for the maximum energy principle and showed that the resulting free energyF=F(T,V,N) is con- sistent with the requirement thatF=U TS andS= ∂F/∂T. − − An analogous consistency check as done for the canonical ensemble can be performed also for the grand canonical ensemble.

Consistency check. For the grand canonical ensemble we haveΩ(T, V, µ) = k BT ln and hence − Z 1 β(Hˆ µNˆ) βΩ β(Hˆ µNˆ) βΩ ˆρ= e− − =e e− − =e − . Z Z WithF=U TS andΩ=F µN we then have − − 1 k ln ˆρ = Ω Hˆ +µ Nˆ − B� � − T −� � � � 1 � � = (F U)=S. − T − Q.e.d.

12.4 Partition function of ideal quantum gases

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

β(Hˆ µNˆ) (T, V, µ) = Tre − − . (12.27) Z Particle number operator. The particle number operator Nˆ entering (12.27) is given by Nˆ= c† c , c † c [0, 1] , a † a =0,1,2,... (12.28) k k � k k� ∈ � k k� �k fermions bosons

wherec k† ck 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 Hˆ is (¯hk2) Hˆ= � c† c ,� = , k= k (12.29) k k k k 2m | | �k for a gas of a non-interacting gas.� k is the dispersion relation. Additive Hamilton operators. The particle number operator Nˆ is always additive. The exponential in the partition function (12.27) factorizes,

β(Hˆ µNˆ) β(� µ)c† c e− − = e− k− k k (12.30) �k when the Hamilton operators is also additive. This holds because particle number oper- ators for different orbitals commute for both fermions and bosons,

† † † † ck1 ck1 , ck2 ck2 = 0, ak1 ak1 , ak2 ak2 = 0, � � � � 154 CHAPTER 12. QUANTUM GASES

wherek =k . 1 � 2 Partition function. We define

ck† ck fermions nk = � � (12.31) � a† a bosons � k k� andfind with ( 12.30) that

βn (� µ) βn (� µ) (T, V, µ) = e− k1 k1 − e− k2 k2 − Z     ··· n n �k1 �k2     βn (� µ) = e− k k− (12.32) � n � �k �k holds for the partition function (T, V, µ) of non-interaction particles. Note that the partition function factorizes onlyZ for the grand canonical partition function.

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

1 (T, V, µ) = (12.33) Z 1 e β(�r µ) r � − − � � − bosons

We have used here an indexr for the sum over orbitals characterized by an energy� r. One may identifyr with the wavevectork whenever needed. Note that ( 12.33) implies that� µ> 0. The chemical potentialµ must hence be small than any eigenenergy� . r − r Fermions. The orbital occupations numbern r takes only the valuesn r = 0 andn r = 1 for a system of fermions. This implies that

β(�r µ) (T, V, µ) = 1 +e − − (12.34) Z r � � � fermions Once has been determined one can obtain the remaining thermodynamic properties of idealZ quantum gases.

12.5 Thermodynamic properties of ideal quantum gases

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

(+) β(�r µ) Ω (T, V, µ) =k T ln 1 e − − (12.35) B − r � � � bosons n † The sum of a geometrical series is ∞ x = 1/(1 x). n=0 − � 12.5. THERMODYNAMIC PROPERTIES OF IDEAL QUANTUM GASES 155

for the bosonic grand canonical potentialΩ (+) = k T ln (+) and − B Z

( ) β(�r µ) Ω − (T, V, µ) = k T ln 1 +e − − (12.36) − B r � � � fermions

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 Nˆ is ∂ 1 Nˆ (+) = Ω(+)(T, V, µ) = , � � − ∂µ eβ(�r µ) 1 r − � − which allows to define the Bose-Einstein distribution

(+) 1 ˆnr = (12.37) � � eβ(�r µ) 1 − − Bose-Einstein

where ˆnr =a k† ak denotes the number operator for the orbitalr. Fermi-Dirac distribution. For Fermions onefinds correspondingly

( ) ∂ ( ) 1 Nˆ − = lnΩ − (T, V, µ) = . � � − ∂µ eβ(�r µ) + 1 r − � The expectation value of the orbital occupation,

( ) 1 ˆnr − = (12.38) � � eβ(�r µ) + 1 − Fermi-Dirac is the Fermi-Dirac distribution function. Thermal equation of state. The Thermal equation of state is

( ) PV= Ω,PV=k T ln ± (T, V, µ). − B Z according to Sect. 5.4, which holds also for ideal quantum gases. ( ) Internal energy. The internal energyU=U ± is given with (5.21) by

∂ ( ) ( ) U µN= βΩ ,U ± = � ˆn ± , − ∂β r � r� r � � � ( ) where the orbital occupation number ˆnr ± is given by (12.37) and (12.38) for bosons and respectively for fermions. The internal� � energy reduces hence to the sum over the mean orbital energies, For the derivation of this intuitive results we consider the case of bosons:

(+) ˆ (+) ∂ (+) �r µ (+) U µ N = ln (T, V, µ) = − = nr (�r µ). − � � − ∂β Z eβ(�r µ) 1 − r − r � − � 156 CHAPTER 12. QUANTUM GASES

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

MB β(�r µ) BE 1 FD 1 n (�r) =e − − , n (�r) = , n (�r) = eβ(�r µ) 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> k BT , both the Bose-Einstein and the Fermi-Dirac distribution functions converge− to� the Maxwell-Boltzmann distribution function.

All three distribution functions are function only ofx=β(� µ). r − Chemical potential – bosons. For bosons the lowest energy state� 0 sets an upper limit for the chemical potential. The occupation

BE 1 n0 (�0) = ,� 0 � r eβ(�0 µ) 1 ≤ − − of the lowest energy state would be otherwise negative, i.e. ifµ would be larger than� 0. Bose-Einstein condensation. There are cases when the chemical potentialµ ap- proaches the lowest energy level� 0 fro below. The occupation of the lowest energy level then diverges: BE 1 lim n0 (�0) = lim β(� µ) . (12.39) µ �0 µ �0 e 0 1 → ∞ → → − − This phenomenon is called Bose-Einstein condensation. –Afinite fractionαN, 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 occupy the 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 the number of particles, 0 n FD 1 (Pauli principle). ≤ ≤ µ can have any value. TheT 0 limit ofn FD is a step function:→

1 1� r < µ lim β(� µ) = T 0 e r + 1 0� r > µ → − � �F = limT 0 µ(T ) is the Fermi energy. → 12.5.2 Particle numberfluctuations

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

1 nrxr nrxr ˆn = n e− , = e− , x =β(� µ). � r� r Z r r − n n Z �r �r We leave the allowed values ofn r, which depend on the statistics, open. We have

∂ 2 1 2 nrxr ˆn = ln , ˆn = n e− , � r� − ∂x Z � r� r r n Z �r where the second relation defines the second moment ˆn2 of the particle number operator � r� ˆnr We evaluate

∂ ∂ 1 nrxr ˆn = n e− ∂x � r� ∂x r r r n Z �r

1 nrxr 1 nrxr 1 2 nrxr = n e− n e− n e− . r r − r � n � � n � n Z �r Z �r Z �r Particle numberfluctuations. Using above expressions wefind that the relation

2 2 ∂ ˆnr ˆnr = ˆnr (12.40) � � − � � − ∂xr � � holds independently of the statistics considered. The variance of the particle number operator is proportional to the (negative) derivative the its expectation value. Relative particle numberfluctuations. Carrying the derivation on the right-hand side of (12.40) out for the our three statistics (MB, BE, FD), we obtain

2 2 2 2 ˆnr ˆnr β(�r µ) ˆnr ˆnr 1 � � − � � =e − , � � − � � = c ˆn 2 ˆn 2 ˆn − � r� � r� � r� 158 CHAPTER 12. QUANTUM GASES for the relative particle numberfluctuations . cMB = 0 for classical particles (Maxwell-Boltzmann) cBE = 1 for bosons (Bose-Einstein) cFD =− +1 for fermions (Fermi-Dirac) The particlefluctuations for bosons are larger than those for classical particles while for fermions they are smaller. This result reflect the intuition that The reason for that is that the Pauli principle hinders the change of a particle state for the case of fermions but not for bosons.