Chapter 12. Quantum gases Microcanonical ensemble

Transcription

1 Chapter 2 Quantu gases In classical statistical echanics, we evaluated therodynaic relations often for an ideal gas, which approxiates a real gas in the highly diluted liit. An iportant difference between classical and quantu echanical any-body systes lies in the distinguishable character of their constituent particles, which gives rise to phenoena and concepts that are not present in classical physics. We start with a reinder of the basic notation. 2. Microcanonical enseble We consider an isolated syste in therodynaical equilibriu, with N particles and energy between E and E + Δ. Coon eigenstates. An isolated enseble in theral equilibriu is characterized by a stationary distribution, i.e. by [ˆρ, Ĥ] = 0, which iplies that the eigenstates of Ĥ Ĥ E n = E n E n, E n E = δ n, E n Ĥ E = E δ n are also eigenstates of ˆρ: E n ˆρ E δ n. Principle of unifor a priori probabilities. For an isolated syste, the principle of a priori probability, as postulated in Sect. 7.5, states that all possible states of the syste have the sae probability. Therefore, we can write: ˆρ icro = P E E Trˆρ =, (2.) with P = { const. E < E < E + Δ 0 otherwise 4

2 42 CHAPTER 2. QUANTUM GASES where the constant is given by the noralization condition Trˆρ =. Energy shell. The operator ˆP Δ = E E, ˆPΔ Ψ = E E Ψ, (2.2) E<E <E+Δ E<E <E+Δ projects to an arbitrary states to the energy shell. The trace Γ(E) of ˆP Δ, ( ) Γ(E) = Tr ˆPΔ = E<E <E+Δ = nuber of states with energies between E and E + Δ, (2.3) is therefore the icrocanonical phase-space volue. Microcanonical density operator. The icrocanonical density operator (2.) is then ˆρ icro = E<E <E+Δ P E E, P = Γ(E), (2.4) with the expectation value of an operator ˆB given in the icrocanonical enseble given by ( ) ˆB = Γ(E) Tr E E ˆB E<E <E+Δ Internal energy. The internal energy U is siply the expectation value of the energy, U = H = E, which one can show alternatively via U = ( ) Γ(E) Tr E E Ĥ E<E <E+Δ E E. Γ(E) E<E <E+Δ Entropy postulate. One possibility to ground quantu statistics is to postulated that the expression for the icrocanonical entropy is within given by S = k B ln Γ(E). (2.5) This definition is the quantu echanical analogu of the postulate (8.5) of the entropy within the classical icrocanonical enseble. There is no Gibbs paradox when the the definition (2.3) for the phase space volue Γ(E) is used. Quantu echanics takes the indistinguishability of identical particles correctly into account.

3 2.. MICROCANONICAL ENSEMBLE 43 Instead of postulating (2.5) one can use the inforation-theoretical expression of the entropy and postulate, in accordance with Sect. 5.5., that the entropy is axial. This point of view will be touched upon Sect Phase space volue. In analogy with classical statistics, we can define the phase space volue Φ(E) as Φ(E) =, E E which easures the nuber of eigenstates of the Hailtonian operator with energies sall or equal to E. The relation between the phase space density Γ(E), the phase space volue Φ(E) and the density of states Ω(E) is Γ(E) = Φ(E + Δ) Φ(E), Ω(E) = δφ δe. For sall Δ0, the entropy (2.5) reduces to S = k B ln(ω(e)δ). 2.. Maxiu entropy principle We ay ground quantu statistic on inforation theoretical principles. One parts fro the Shannon entropy I[P ], I[P ] = P ln P, (2.6) introduced in Sect , deanding in a second step that the diensionless entropy, S/k B corresponds to a easure of the disorder present in the syste and that this easure is quantified by I[P ]. Maxial entropy postulate. In Sect we have noted that the second law of therodynaics iplies that axiu entropy principle, which states that irreversible processes ay only increase the entropy in isolated systes. Reversing the stance we ay use the axiu energy principle as the basic postulate on which to ground quantu statistics. POSTULATE A statistical enseble is therally equilibrated when the Shannon entropy (2.6) is axial. This approach coes with the positive side effect that is holds for all three ensebles. Maxial entropy distributions. In order to put the axiu entropy postulate at work we need to know which are the axial entropy distributions. For the case of the icrocanonical enseble we need therefore to know which function p(y) axiizes I[p] = p(y) ln ( p(y) ) dy, δi = [ln ( ) ] p(y) + δp(y) dy, (2.7) }{{} 0

4 44 CHAPTER 2. QUANTUM GASES where we have denoted the variation of I[p] with δi. With the variation δp(y) of the distribution function being arbitrary one has that the bracket on the right-hand-side of δi in (2.7) needs to vanish for δi = 0. It follows that p(y) = const.. (2.8) Microcanonical entropy. The unifor distribution (2.8) reflects the principle of unifor a priori probabilities entering the forulation of the icrocanonical density atrix (2.4). for the entropy, S = k B E<E <E+Delta P ln(p ), P = Γ(E), it then follows S = k ( ) B Γ(E) ln, S = k B ln Γ(E), (2.9) Γ(E) Γ(E) which reproduces the energy postulate (2.5). Γ(E) is here the volue of the energy shell Third law of therodynaics The third law of therodynaics is of quantu echanical nature: The entropy of a therodynaical syste at T = 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 syste with a discrete energy spectru, there is a lowest energy state, the ground state. At T 0, the syste will go into this state. If the groundstate is n-ties degenerate, the entropy of the syste at T = 0 is n: degeneracy ultiplicity. For n = (no degeneracy), S = 0. S(T = 0) = k B ln n, Spontaneous syetry breaking. The entropy S apparently doesn t fulfill the third law of therodynaics for a finite ground state degeneracy n >. However, there is no paradox. One uses variational calculus to derive Newton s equations of otions, d L dt q = L q, W (L) = L (q, q) dt, fro the action W (L).

5 2.. MICROCANONICAL ENSEMBLE 45 When n >, the groundstate is degenerate due to the existence of internal syetries of the Hailtonian (for instance, spin rotational syetry); at T = 0 the syetry gets broken through a phase transition that lets the entropy go to zero. Spin rotational syetry. The agnetization can point in an arbitrary direction for a agnetic copound with spin rotational syetry. In this case, one of the any possible degenerate states is spontaneously selected Two-level syste and the concept of negative teperature Consider an isolated syste of N spin-/2 particles, which have each two spin orientations, s = /2 and s z = ±/2. We assue that the two possible energy states ±ɛ are occupied by N ± particles, with N = N + + N, E = (N + N ) ɛ, where E is the total energy. Obviously, N + = ( N + E 2 ɛ ), N = 2 ( N E ) ɛ. (2.0) Volue of the energy shell. The nuber of possible states with energy E and particle nuber N is given by the density of states Ω(E, N): ( ) N N! Γ(E, N) = = N +! N!, N + as there are ( N N + ) possibilities to select N+ particles out of a total of N indistinguishable particles. Microcanonical entropy. The entropy is then ( ) N! S(E, N) = k b ln Γ(E, N) = k B ln N +!N! [ ] = k B ln(n!) ln(n +!) ln(n!) [ ] k B N ln N N + ln N + N ln N, (2.) where we have ade use of the Stirling forula ln(n!) N ln N in the last step. Teperature. The teperature of the collection of two-level is then T = S ( ) N+ ( ) E = k B ln N + + N E k N B ln N + E = k B ( ln N+ ln N ) /2ɛ,

6 46 CHAPTER 2. QUANTUM GASES where we have used (2.), naely that N ± / E = ±/(2ɛ). We then have T = k B 2ɛ ln ( N N + ), N + N = e β2ɛ, 2N ± = N ± E/ɛ. (2.2) N > N + The noral state is characterized by E < 0, i.e. by a situation where there are ore particles in the lower energy level ɛ than in the higher energy level ɛ > 0. The teperature is then positive, T > 0, as usual. All particles are in the lower energy level for N = N, which corresponds to T = 0 and β. The slope /T = S/ E diverges at E = ɛ. 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 (2.) is then S(E = 0, N) = k B N [ ln N ln(n/2) ] = k B N ln 2. The entropy per particle is given at T by the logarith of the nuber of degrees of freedo (in units of k B ). The slope /T = S/ E vanishes at E = 0. The teperature T jups fro + to as soon as ore 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 reain in the excited state for a certain

7 2.2. CANONICAL ENSEMBLE 47 period of tie (viz if the excited state is etastable), the situation N + > N is realized. This phenoenon is called inversion. The definition (2.2) of the teperature, T = k ( ) B 2ɛ ln N, N + becoes forally negative whenever N + > N. The concept of negative teperature akes only sense for systes with an upper bound for the energy. The inversion of the level occupation needs external puping. Negative teperatures do therefore not occur for isolated systes. There is no true negative teperature. T 0 fro below when all the particles are in the higher level. 2.2 Canonical enseble In order to drive the density atrix ˆρ for the canonical enseble one follow the route taken in Sect. 9. for the classical enseble. Here we will start, alternatively, fro the axiu entropy postulate introduce in Sect. 2.. as a possible grounding principle for statistical echanics. Lagrange paraeters. We start by deterining the probability distribution p(y) axiizing the Shannon inforation I[p] under the constraint of a given expectation value y. This is done within variational calculus by finding the axiu of [ I[p] λ y = ln ( p(y) ) ] + λ y p(y) dy, (2.3) where λ is an appropriate Lagrange paraeter. Maxiu entropy distribution. Variation of (2.3) yields δ [ I[p] λ y ] [ = + ln ( p(y) ) ] + λ y δp(y) dy. } {{ } 0 Copare the equivalent expression (2.7) without constraints. We therefore find that p(y) e λy, with the noralization factor. p(y) = e λy, = dy e λy, (2.4)

8 48 CHAPTER 2. QUANTUM GASES Syste in theral contact with a heat bath. The energy fluctuates for a syste A in theral equilibriu with a reservoir A 2, which is the setup of the canonical enseble. We therefore need to find the distribution function P axiizing S λ E = k B P ln(p ) k B λ E P, where we rescaled the Lagrange paraeter λ by k B. Our result (2.4) then tells us, that P = e λe, = e λe. (2.5) What reains to be done is to deterine the physical significance of the λ. Teperature as a Lagrange paraeter. We will now show that λ = k B T, U = E e βe, = e βe, (2.6) which iplies that the (inverse) teperature β can be defined as the Lagrange paraeter needed to regulate the agnitude of the internal energy U = E. Free energy. The relation (2.6) is verified by showing that there exist for λ = β a function F fulfilling with F T = S k B P ln(p ), F = U T S. (2.7) the defining therodynaic relations of the free energy. We use ln(p ) = λe ln, which allows to rewrite the entropy as T S = k B T P ( βe + ln ) = λβ E + k }{{}}{{} B T ln. (2.8) }{{} = = U F Defining the free energy as F = k B T ln is therefore consistent with F = U T S. The first condition of (2.7) is verified by evaluating F T = [ kb T ln ] = k B ln + k BT T = k B ln + T E S, where we copared with (2.8) in the last step. ( ) e βe ( E ) k B T 2

9 2.2. CANONICAL ENSEMBLE 49 Specific heat. With (4.8) and (2.8), naely ( ) S = C V T T, S = k B ln + T V,N E e βe we obtain ( ) S T V,N = k ( ) ( ) B e βe ( E )e βe + E k B T 2 T 2 + ( ) ( E 2 T ) e βe + E ( ) ( E k B T 2 ) e βe T k B T 2, which leads to C V = k b T 2 [ E 2 E 2 ]. (2.9) The quantuechanical result (2.9) for the specific heat C V is is hence identical to the classical expression (9.23). Canonical density atrix. Using that the eigenstates E of Hailton operator Ĥ for a coplete basis, Ĥ E = E E, E E = ˆ, we write the canonical density atrix operator ˆρ as ˆρ = e βe E E = e βĥ E E } {{ } ˆ, that is as ˆρ = e βĥ, = Tr [ e βĥ], (2.20) where we have used that the canonical partition function as = (T, V, N) = e βe Quantu echanical haronic oscillators As an application we consider N localized independent linear haronic oscillators with frequency ω in theral equilibriu at teperature T. Such a syste would describe, for instance, the radiation energy of a black body. Spectru. A single oscillator is described by ĤΨ n = ε n Ψ n, ε n = hω ( n + ), 2 where n is the occupation nuber. The spectru ɛ n is equally spaced.

10 50 CHAPTER 2. QUANTUM GASES Partition function for one oscillator. Then, the partition function in the canonical enseble for one oscillator is given by (T, V, N = ) = Tre βĥ = e βεn = n ( = e ) β hω 2 e β hω n, which is a geoetrical series of the for Therefore, (T, V, ) = n=0 n=0 r n = r. β hω e 2 e = β hω e β hω 2 e β hω 2 = e β hω(n+ 2) n=0 2 sinh ( β hω 2 Partition function for N oscillators. For N particles it then follows that (T, V, N) = [ (T, V, ) ] ( )] N N β hω = [2 sinh. 2 The above expression is valid only because we assued that the oscillators don t interact with each other and that all eigenfrequencies are idential. Free energy. The free energy F = k B T ln() is ( ) e β hω/2 F (T, V, N) = Nk B T ln e β hω For the derivatives we obtain ( ) F µ = N = T,V N 2 hω }{{} zero-point energy of N oscillators ). +Nk B T ln ( e β hω). = F ( ) F N, P = V T,N = 0. Note that the pressure P vanishes due to the fact that the oscillators are localized in the sense that the eigenfrequencies ω do not depend on the volue V. Entropy. The entropy is ( ) F S = T V,N = Nk B ln ( e β hω) ( ) hωe β hω Nk B T e β hω k B T 2 [ β hω = Nk B e β hω ln ( e β hω) ].

11 2.3. GRAND CANONICAL ENSEMBLE 5 Internal energy. The internal energy U = F + T S is [ ] U = N hω 2 +, (2.2) e β hω which can be written as U = N hω ( ) 2 + n, n = e β hω, where n is the particle occupation nuber. We will see thus expression again when we introduce the Bose gas. The T 0 liit, U N 2 hω, gives the ground-state energy of a syste of N quantu oscillators (vacuu fluctuations). For T one recovers with U Nk B T the expression for the energy N classical oscillators. 2.3 Grand canonical enseble A grand canonical enseble ay exchange both energy and particles with a reservoir. The eigenstates depend therefor on the nuber N of particles present: Ĥ E (N) = E (N) E (N), ˆN E (N) = N E (N), ˆ = N, E (N) E (N), where ˆN is the particle nuber operator. Maxiu entropy postulate. In order to apply the axiu entropy principle to the grand canonical enseble we have therefore to find the distribution function P (N) axiizing the Shannon entropy under the constraints that the expectation values of the energy and of the particle nuber, E = Ĥ, N = ˆN, are given. Following the calculation perfored in Sect. 2.2 for the canonical enseble we look for the distribution P (N) axiizing [ k B I[P ] λe Ĥ λ N ˆN ] [ = k B P (N) ln P (N) + λ E E (N) + λ N N ], N,

12 52 CHAPTER 2. QUANTUM GASES where I[P ] is the Shannon entropy. The solution is, in analogy to (2.5), P (N) = e λ EE λ N N, = e λ EE λ N N. (2.22) N, The cheical potential as an Lagrange paraeter. The physical significance of the two Lagrange paraeters involved are given by λ E β, λ N βµ. (2.23) The first equivalence was shown in Sect The second ter in (2.23) iplies that the cheical potential µ ay be defined, odulo a factor ( β), a the Lagrange paraeter regulating the particle nuber. Density atrix. Using (2.23) we find ˆρ = e β(ĥ µ ˆN), = Tr [ e β(ĥ µ ˆN) ] (2.24) for the grand canonical density atrix ˆρ and respectively for the grand canonical potential Ω. Ω(T, V, µ) = k B T ln (T, V, µ) (2.25) Particle nuber. In order to verify (2.23) we confir that ( ) ( ) Ω N = = k B T µ T,V µ ln e β(e(n) µn) N, }{{} β( N, Ne β(e(n) µn) )/ = β ˆN holds Entropy and the density atrix operator We showed in Sect. 2.. that the entropy S can be written within the icrocanonical enseble in ters of the density atrix operator ˆρ as S = k B Tr(ˆρ ln ˆρ) = k B ln ˆρ, (2.26) i.e., the entropy is given by the expectation value of the negative logarith of the density atrix operator ˆρ. Note that since the eigenvalues of ˆρ are probabilities, i.e., they are between 0 and, the logarith in Eq. (2.26) will therefore be negative and S positive.

13 2.4. PARTITION FUNCTION OF IDEAL QUANTUM GASES 53 For the canonical enseble we used (2.2) as the starting point for the axiu energy principle and showed that the resulting free energy F = F (T, V, N) is consistent with the requireent that F = U T S and S = F/ T. An analogous consistency check as done for the canonical enseble can be perfored also for the grand canonical enseble. Consistency check. For the grand canonical enseble we have Ω(T, V, µ) = k B T ln and hence ˆρ = e β(ĥ µ ˆN) = e βω β(ĥ µ ˆN) e = e βω. With F = U T S and Ω = F µn we then have k B ln ˆρ = ( ) Ω Ĥ + µ ˆN T Q.e.d. = (F U) = S. T 2.4 Partition function of ideal quantu gases The adequate enseble to deal with quantu gases is the grand canonical enseble: (T, V, µ) = Tr e β(ĥ µ ˆN). (2.27) Particle nuber operator. The particle nuber operator ˆN entering (2.27) is given by ˆN = c k c k, c k c k [0, ], a k }{{} a k = 0,, 2,... (2.28) }{{} k ferions bosons where c k c k is the operator easuring the occupancy of a plane wave with oentu hk. Two ferions ay never occupy the sae orbital. Hailtonian. The Hailton operator Ĥ is Ĥ = k ɛ k c k c k, ɛ k = ( hk)2, k = k (2.29) 2 for a gas of a non-interacting gas. ɛ k is the dispersion relation. Additive Hailton operators. The particle nuber operator ˆN is always additive. The exponential in the partition function (2.27) factorizes, e β(ĥ µ ˆN) = k e β(ɛ k µ)c k c k (2.30) when the Hailton operators is also additive. This holds because particle nuber operators for different orbitals coute for both ferions and bosons, [ ] [ ] c k c k, c k 2 c k2 = 0, a k a k, a k 2 a k2 = 0,

14 54 CHAPTER 2. QUANTUM GASES where k k 2. Partition function. We define n k = { c k c k ferions a k a k bosons (2.3) and find with (2.30) that (T, V, µ) = nk e βn k (ɛ k µ) nk2 e βn k 2 (ɛ k2 µ) = k ( n k e βnk(ɛk µ) ) (2.32) holds for the partition function (T, V, µ) of non-interaction particles. Note that the partition function factorizes only for the grand canonical partition function. Bosons. The occupation nubers n r runs over all non-negative integers for a syste of bosons, so that the parenthesis of (2.32) reduces to a geoetrical series : (T, V, µ) = r [ ] e β(ɛr µ) bosons (2.33) We have used here an index r for the su over orbitals characterized by an energy ɛ r. One ay identify r with the wavevector k whenever needed. Note that (2.33) iplies that ɛ r µ > 0. The cheical potential µ ust hence be sall than any eigenenergy ɛ r. Ferions. The orbital occupations nuber n r takes only the values n r = 0 and n r = for a syste of ferions. This iplies that (T, V, µ) = r [ + e β(ɛ r µ) ] ferions (2.34) Once has been deterined one can obtain the reaining therodynaic properties of ideal quantu gases. 2.5 Therodynaic properties of ideal quantu gases Using the index (+) and ( ) for bosons and ferions respectively we find with (2.33) Ω (+) (T, V, µ) = k B T r ln [ e β(ɛr µ)] The su of a geoetrical series is n=0 xn = /( x). bosons (2.35)

15 2.5. THERMODYNAMIC PROPERTIES OF IDEAL QUANTUM GASES 55 for the bosonic grand canonical potential Ω (+) = k B T ln (+) and Ω ( ) (T, V, µ) = k B T r ln [ + e β(ɛr µ)] ferions for the case of ferions. The volue dependence results fro the su r (2.36) over orbitals. Bose-Einstein distribution. The expectation value of the particle nuber operator ˆN is ˆN (+) = µ Ω(+) (T, V, µ) = e β(ɛr µ), r which allows to define the Bose-Einstein distribution ˆn r (+) = e β(ɛr µ) Bose-Einstein (2.37) where ˆn r = a k a k denotes the nuber operator for the orbital r. Feri-Dirac distribution. For Ferions one finds correspondingly ˆN ( ) = µ ln Ω( ) (T, V, µ) = r e β(ɛr µ) +. The expectation value of the orbital occupation, ˆn r ( ) = is the Feri-Dirac distribution function. e β(ɛr µ) + Feri-Dirac Theral equation of state. The Theral equation of state is (2.38) P V = Ω, P V = k B T ln (±) (T, V, µ). according to Sect. 5.4, which holds also for ideal quantu gases. Internal energy. The internal energy U = U (±) is given with (5.2) by U µn = ( ) βω, U (±) = β r ɛ r ˆn r (±), where the orbital occupation nuber ˆn r (±) is given by (2.37) and (2.38) for bosons and respectively for ferions. The internal energy reduces hence to the su over the ean orbital energies, For the derivation of this intuitive results we consider the case of bosons: U (+) µ ˆN (+) = β ln (+) (T, V, µ) = r ɛ r µ e β(ɛr µ) = r n (+) r (ɛ r µ).

16 56 CHAPTER 2. QUANTUM GASES 2.5. Maxwell-Boltzann, Bose-Einstein and Feri-Dirac distribution The three distributions n MB (ɛ r ) = e β(ɛr µ), n BE (ɛ r ) = e β(ɛr µ), nf D (ɛ r ) = quantify the occupation of an energy level for a non-interacting gas. n MB : Maxwell-Boltzann (classical grand canonical enseble) n BE : Bose-Einstein (quantu) n F D : Feri-Dirac (quantu) e β(ɛr µ) + For large one-particle energies, i.e., for ɛ r µ > k B T, both the Bose-Einstein and the Feri-Dirac distribution functions converge to the Maxwell-Boltzann distribution function. All three distribution functions are function only of x = β(ɛ r µ). Cheical potential bosons. For bosons the lowest energy state ɛ 0 sets an upper liit for the cheical potential. The occupation n BE 0 (ɛ 0 ) = e β(ɛ 0 µ), ɛ 0 ɛ r of the lowest energy state would be otherwise negative, i.e. if µ would be larger than ɛ 0. Bose-Einstein condensation. There are cases when the cheical potential µ approaches the lowest energy level ɛ 0 fro below. The occupation of the lowest energy level then diverges: li n BE 0 (ɛ 0 ) = li µ ɛ 0 µ ɛ0 e β(ɛ 0 µ). (2.39) This phenoenon is called Bose-Einstein condensation. A finite fraction αn, with 0 α, of all particles occupy the identical state, naely ɛ 0 when (2.39) occurs. The reaining fraction ( α)n of particles occupy the higher level. 4 He is a faous exaple of a real gas undergoing a Bose-Einstein condensation. The residual interactions between the Heliu atos are however substantial.

17 2.5. THERMODYNAMIC PROPERTIES OF IDEAL QUANTUM GASES 57 Cheical potential ferions. The cheical potential regulates the nuber of particles, 0 n F D (Pauli principle). µ can have any value. The T 0 liit of n F D is a step function: { li T 0 e β(ɛr µ) + = ɛr < µ 0 ɛ r > µ ɛ F = li T 0 µ(t ) is the Feri energy Particle nuber fluctuations We restrict ourselves to a single orbital with orbital energy ɛ r and write the expectation value of the particle nuber as ˆn r = n r e nrxr, = e nrxr, x r = β(ɛ r µ). n r n r We leave the allowed values of n r, which depend on the statistics, open. We have ˆn r = x r ln, ˆn 2 r = n r n 2 r e nrxr, where the second relation defines the second oent ˆn 2 r of the particle nuber operator ˆn r We evaluate ˆn r = n r e nrxr x r x r n ( r = n r e nrxr n r ) ( ) n r e nrxr n 2 r e nrxr. n r Particle nuber fluctuations. Using above expressions we find that the relation n r ˆn 2 r ˆn r 2 = x r ˆn r (2.40) holds independently of the statistics considered. The variance of the particle nuber operator is proportional to the (negative) derivative the its expectation value. Relative particle nuber fluctuations. Carrying the derivation on the right-hand side of (2.40) out for the our three statistics (MB, BE, FD), we obtain ˆn 2 r ˆn r 2 ˆn r 2 = e β(ɛr µ), ˆn 2 r ˆn r 2 ˆn r 2 = ˆn r c

18 58 CHAPTER 2. QUANTUM GASES for the relative particle nuber fluctuations. c MB = 0 for classical particles (Maxwell-Boltzann) c BE = for bosons (Bose-Einstein) c F D = + for ferions (Feri-Dirac) The particle fluctuations for bosons are larger than those for classical particles while for ferions they are saller. 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 ferions but not for bosons.