Quantum Grand Canonical Ensemble

Chapter 16

How do we proceed quantum mechanically? For fermions the wavefunction is antisymmetric. An N particle basis function can be constructed in terms of single-particle wavefunctions as follows: 1 ψ(r , r ,..., r )= ( 1)P P φ (r )φ (r ) φ (r ), (16.1) 1 2 N √ − 1 1 2 2 · · · N N N! XP where P is the permutation operator, and ( 1)P = 1 depending on whether P is an even or odd permutation. This result− can± be written as a (Slater) determinant 1 ψ(r1, r2,..., rN )= det φi(rj ). (16.2) √N! For bosons, the wavefunction must be symmetric: 1 1 ψ(r1, r2,..., rN )= P φ1(r1) φN (rN ), (16.3) √N1! Nl! √N! · · · · · · XP where Nj is the number of occurrences of φj . The extra combinatorial factor comes from the fact that you get a distinct wavefunction N !N ! N ! times. 1 2 · · · l A subsystem consists of Nj particles, with total energy Ej . It is described by a state vector Ej ,Nj , kj , where kj are the other quantum numbers necessary to specify the| state. Thei system consists of subsystems which do not interact with each other. For fermions, the system is described by a state vector

n P 1 E,N,k = ( 1) P Ej ,Nj , kj Nj !, (16.4) | i − √N! | i XP jY=1 p where the outer product is over vectors in diﬀerent Hilbert spaces, and P per- mutes particle between the diﬀerent spaces. For bosons, the ( 1)P factor would not be present. −

89 Version of April 26, 2010 90 Version of April 26, 2010CHAPTER 16. QUANTUM GRAND CANONICAL ENSEMBLE

The microcanonical distribution for the entire system is described by the density operator

E+ǫ/2 1 ′ ′ ρǫ(E) = dE δ(E H) Ωǫ(E,N) ZE−ǫ/2 − 1 = E,N,k E,N,k , (16.5) Ωǫ(E,N) | ih | Xk where the sum is over all states having energies beween E ǫ/2 and E + ǫ/2. Here, as before, energies are measured in units of ǫ, so all− these energies are considered the same. The averaged structure function is

Ω (E,N)= E,N,k E,N,k , (16.6) ǫ h | i Xk which is the number of states in the energy interval [E ǫ/2, E + ǫ/2] and with occupation number N. The total degeneracy of the entire− system is again given by the convolution law

Ω (E,N)= δ δ Ω (E ,N ). (16.7) ǫ E, Ej N, Nj ǫj j j j j X Yj {Nj }{Ej } P P Note that there are no N!s because they are included in the deﬁnitions of the physical states. The single-subsystem distribution function is

(n−1) (1) Ωǫ (E E1,N N1) E1,N1 = (n−) − , (16.8) P Ωǫ (E,N) which is the ratio of the number of states for which subsystem 1 has energy E1 and occupation number N1 to the total number of states. Again, there are no N!s. The grand structure function is here deﬁned by ∞ (E,z)= zN Ω (E,N). (16.9) Wǫ ǫ NX=0 For the composite system

(E,z) = δ δ zNj Ω (E ,N ) ǫ E, Ej N, Nj ǫ j j W j j XN X Yj XNj {Ej } P P = δ (E ,z). (16.10) E, Ej ǫj j j W X Yj {Ej } P The grand partition function is

(α, z) = e−αE (E,z)= zN χ(α, N) Xǫ Wǫ XE XN = (α, z). (16.11) Xǫj Yj 91 Version of April 26, 2010

Once again, we do an asymptotic evaluation using

′ ǫ α(E−e ) δE,E′ = dα e , (16.12) 2πi ZC from which we deduce e−β(E1−µN1) (1) = , (16.13) PE1,N1 (β, µ) X1 and in turn, by recognizing E1, N1 as the eigenvalues of the Hamiltonian and number operator for the single subsystem, we deduce the density operator

e−β(H−µN ) ρ = , (16.14) (β, µ) X where H is the Hamiltonian operator, whose eigenvalues are the possible energy states of the system, and is the number operator, whose eigenvalues are the number of particles inN the system. Again note, in contradistinction with the classical probability distribution, there is no N! because the combinatorical factors are taken care of in the deﬁnition of the quantum state vectors. This holds whether the particles are bosons or fermions. Because

Tr ρ =1, (16.15) the grand partition function is

(β, µ) = Tr e−β(H−µN ) X = E,N,k e−β(H−µN ) E,N,k h | | i E,N,kX −β(E−µN) = e gE,N , (16.16) E,NX where gE,N is the degeneracy of the state with energy E and number of particles N. Now ∂ ln (β, µ) = H µ U µN, (16.17) −∂β X h − Ni≡ − 1 ∂ ln (β, µ) = = N, (16.18) β ∂µ X hN i where U and N are the thermodynamic quantities. The pressure is ∂H 1 ∂ p = = = ln (β, µ), (16.19) hFV i −h ∂V i β ∂V X so therefore

d ln (β, µ, V )= (U µN)dβ + βN dµ + βp dV, (16.20) X − − 92 Version of April 26, 2010CHAPTER 16. QUANTUM GRAND CANONICAL ENSEMBLE or from Eq. (14.26)

d[β(U µN) + ln ] = β(dU d(µN)) + βN dµ + βp dV − X − dS = β[dU + p dV µ dN]= βδQ = , (16.21) − k so, up to a constant, S = β(U µN) + ln , (16.22) k − X or TS = U µN + kT ln . (16.23) − X This suggests deﬁning still another kind of free energy, the grand potential,

J = F µN = U TS µN = kT ln , (16.24) − − − − X which is analogous to F = kT ln χ. Note that − dJ = T dS p dV + µ dN d(TS) d(µN) − − − = p dV S dT N dµ, (16.25) − − − which says that J(T, µ, V ) is a function of the indicated variables, that is,

∂J ∂J ∂J = S, = N, = p. (16.26) ∂T µ,V − ∂µ T,V − ∂V T,µ −

The last two equations are just those given in Eqs. (16.19) and (16.18), while the last is ∂J ∂ = k ln kT ln ∂T − X − ∂T X 1 ∂ = k ln + ln − X T ∂β X 1 = [ U + µN kT ln ]= S. (16.27) T − − X −

16.1 Bose-Einstein and Fermi-Dirac Distributions

The grand structure function for a gas of noninteracting particles is (no N!)

= zN χ(α, N), (16.28) X XN where χ(α, N)= δ e−βnj εj , (16.29) N, nj j X Yj {nj } P 16.1. BOSE-EINSTEIN AND FERMI-DIRAC DISTRIBUTIONS93 Version of April 26, 2010

where nj is the number of particles in the single-particle energy state εj. Thus

= znj e−βnj εj X {Xnj } Yj = eβ[µnj −nj εj ], (16.30)

Yj Xnj which also immediately follows from

−β nj εj +βµ nj = e−β(E−µN) = e j j . (16.31) X P P N,EX {Xnj }

For particles obeying Bose-Einstein statistics, bosons, the sum on nj ranges from 0 to , so ∞ ∞ 1 znj e−βnj εj = , (16.32) 1 ze−βεj nXy =0 − while for particles obeying Fermi-Dirac statistics, fermions, the sum on nj ranges only from 0 to 1: 1 znj e−βnj εj =1+ ze−βεj . (16.33)

nXj =0 so in general znj e−βnj εj = (1 ze−βεj )±1, (16.34) ± Xnj where the upper sign refers to fermions, and the lower to bosons. Thus the grand partition function is

= (1 ze−βεj )±1, (16.35) X ± Yj and ln = ln(1 ze−βεj )= ln(1 eβµe−βεj ). (16.36) X ± ± ± ± Xj Xj Then, the total number of particles is

1 ∂ eβ(µ−εj ) N = ln = β ∂µ X 1 eβ(µ−εj ) Xj ± 1 = , (16.37) eβ(εj −µ) 1 Xj ± and ∂ ε µ U µN = ln = j − , (16.38) − −∂β X eβ(εj −µ) 1 Xj ± 94 Version of April 26, 2010CHAPTER 16. QUANTUM GRAND CANONICAL ENSEMBLE which implies that the thermodynamic energy is ε U = j . (16.39) eβ(εj −µ) 1 Xj ± The mean number of particles in the l energy level is 1 ∂ eβ(µ−εl) nl = ln = β(µ−ε ) h i −β ∂εl X 1 e l ± 1 = , (16.40) eβ(εl−µ) 1 ± so as expected N = n , U = n ε . (16.41) h li h li l Xl Xl βµ These results coincide with those found in Sec. 12.1, with ζ0 = e .

16.2 Photons

For photons, there is no restriction on the number of particles, so we can set z =1 or µ = 0: = e−βnj εj = (1 e−βεj )−1, (16.42) X − {Xnj } Yj Yj ∂ ε U = ln = j , (16.43) −∂β X eβεj 1 Xj − 1 ∂ 1 nj = ln = . (16.44) h i −β ∂ε X eβεj 1 j − The ﬂuctuation in the individual level occupation numbers is (n n )2 = n2 n 2 h l − h li i h l i − h li 1 1 ∂2 1 1 ∂ 2 = β2 ∂ε2 X − β2 ∂ε X X l X l 1 ∂2 1 ∂ = 2 2 ln = nl β ∂εl X −β ∂εl h i βεl e 2 = = nl + nl (eβεl 1)2 h i h i − = n (1 + n ). (16.45) h li h li 16.3 Planck Distribution

For a photon gas in a volume V , the number of states in a wavenumber interval (dk) is 2 V (dk) , ¯hk = p, E = pc, (16.46) (2π)3 16.3. PLANCK DISTRIBUTION 95 Version of April 26, 2010 where the factor of 2 emerges because photons are helicity 2 particles; that is, there are two polarization states for each momentum state. Then the logarithm of the grand partition function is

ln = ln 1 e−βεj X − − Xj 8πV ∞ = dk k2 ln 1 e−βhck¯ −(2π)3 Z − 0 V 1 ∞ 1 ∞ e−βhck¯ = k3 ln 1 e−βhck¯ dk k3 β¯hc −π2 3 − − 3 Z 1 e−βhkc¯ 0 0 − Vβ¯hc ∞ 1 F = dk k3 = , (16.47) 3π2 Z eβhck¯ 1 −kT 0 − where F is either the Helmholtz free energy or the grand potential (the distinc- tion disappears when µ = 0). The internal energy is ∂ ε V ∞ dk k3 U = ln = l = ¯hc . (16.48) −∂β X eβεl 1 π2 Z eβhck¯ 1 Xl − 0 − We see here the characteristic Planck distribution. In the classical limit,h ¯ 0, → ∞ V 2 V 2 kT dk k . (16.49) → π Z0 which exhibits the Rayleigh-Jeans law, and exhibits the famous ultraviolet catas- trophe. This breakdown of classical physics led Planck to the introduction of the quantum of light, the photon. Note that here 1 F = U, (16.50) −3 and so the pressure is ∂F 1 U p = = , (16.51) −∂V 3 V the characteristic law for a radiation gas. To determine the total energy, recall that from Eq. (7.16) ∞ xn−1 dx = Γ(n)ζ(n), (16.52) Z ex 1 0 − where ζ(n) is the Riemann zeta function. For integer argument the zeta function may be expressed as a Bernoulli number, (2π)2n ζ(2n)= B . (16.53) 2(2n)! n In this way we ﬁnd ∞ x3 π4 dx = , (16.54) Z ex 1 15 0 − 96 Version of April 26, 2010CHAPTER 16. QUANTUM GRAND CANONICAL ENSEMBLE and then we recover Stefan’s law: π2 k4 U = 3F = T 4V, (16.55) − 15 (¯hc)3 and the speciﬁc heat for the photon gas,

∂U 4π2 k4 c = = T 3V. (16.56) v ∂T 15 (¯hc)3