Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 24
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The phase factor ei(pa xa −pa ta /2M )/h̄on the right-hand side of Eq. (1.529), which is unity in the limit performed in thatequation, is kept in Eq. (4.540) for later convenience.Formula (1.529) is a reliable starting point for extracting the scattering amplitudefpb pa from the time evolution amplitude in x-space (xb 0|xa ta ) at xa = pa ta /M byextracting the coefficient of the outcoming spherical wave eipa r/h̄ /r.As a cross check we insert the free-particle amplitude (1.338) into (1.529) andobtain the free undisturbed wave function eipa x , which is the correct first term inEq. (1.524) associated with unscattered particles.(D)H. Kleinert, PATH INTEGRALS771.17 Classical and Quantum Statistics1.17Classical and Quantum StatisticsConsider a physical system with a constant number of particles N whose Hamiltonian has no explicit time dependence.
If it is brought in contact with a thermalreservoir at a temperature T and has reached equilibrium, its thermodynamic properties can be obtained through the following rules: At the level of classical mechanics,each volume element in phase spacedp dqdp dq=h2πh̄(1.533)is occupied with a probability proportional to the Boltzmann factore−H(p,q)/kB T ,(1.534)where kB is the Boltzmann constant,kB = 1.3806221(59) × 10−16 erg/Kelvin.(1.535)The number in parentheses indicates the experimental uncertainty of the two digitsin front of it.
The quantity 1/kB T has the dimension of an inverse energy and iscommonly denoted by β. It will be called the inverse temperature, forgetting aboutthe factor kB . In fact, we shall sometimes take T to be measured in energy units kBtimes Kelvin rather than in Kelvin. Then we may drop kB in all formulas.The integral over the Boltzmann factors of all phase space elements,23Zcl (T ) ≡Zdp dq −H(p,q)/kB Te,2πh̄(1.536)is called the classical partition function.
It contains all classical thermodynamicinformation of the system. Of course, for a generalZHamiltonian system with manyYdpn dqn /2πh̄. The reader maydegrees of freedom, the phase space integral isnwonder why an expression containing Planck’s quantum h̄ is called classical . Thereason is that h̄ can really be omitted in calculating any thermodynamic average.In classical statistics it merely supplies us with an irrelevant normalization factorwhich makes Z dimensionless.1.17.1Canonical EnsembleIn quantum statistics, the Hamiltonian is replaced by the operator Ĥ and the integralover phase space by the trace in the Hilbert space.
This leads to the quantumstatistical partition functionZ(T ) ≡ Tr e−Ĥ/kB T ≡ Tr e−H(p̂,x̂)/kB T ,23(1.537)In the sequel we shall always work at a fixed volume V and therefore suppress the argumentV everywhere.781 Fundamentalswhere Tr Ô denotes the trace of the operator Ô. If Ĥ is an N-particle SchrödingerHamiltonian, the quantum-statistical system is referred to as a canonical ensemble.The right-hand side of (1.537) contains the position operator x̂ in Cartesian coordinates rather than q̂ to ensure that the system can be quantized canonically. In casessuch as the spinning top, the trace formula is also valid but the Hilbert space isspanned by the representation states of the angular momentum operators.
In moregeneral Lagrangian systems, the quantization has to be performed differently in theway to be described in Chapters 10 and 8.At this point we make an important observation: The quantum partition function is related in a very simple way to the quantum-mechanical time evolution operator. To emphasize this relation we shall define the trace of this operator fortime-independent Hamiltonians as the quantum-mechanical partition function:ZQM (tb − ta ) ≡ Tr Û(tb , ta ) = Tr e−i(tb −ta )Ĥ/h̄ .(1.538)Obviously the quantum-statistical partition function Z(T ) may be obtained fromthe quantum-mechanical one by continuing the time interval tb − ta to the negativeimaginary valueih̄≡ −ih̄β.(1.539)tb − ta = −kB TThis simple formal relation shows that the trace of the time evolution operatorcontains all information on the thermodynamic equilibrium properties of a quantumsystem.1.17.2Grand-Canonical EnsembleFor systems containing many bodies it is often convenient to study their equilibriumproperties in contact with a particle reservoir characterized by a chemical potentialµ.
For this one defines what is called the grand-canonical quantum-statistical partition functionZG (T, µ) = Tr e−(Ĥ−µN̂ )/kB T .(1.540)Here N̂ is the operator counting the number of particles in each state of the ensemble.The combination of operators in the exponent,ĤG = Ĥ − µN̂ ,(1.541)is called the grand-canonical Hamiltonian.Given a partition function Z(T ) at a fixed particle number N, the free energy isdefined byF (T ) = −kB T log Z(T ).(1.542)Its grand-canonical version at a fixed chemical potential isFG (T, µ) = −kB T log ZG (T, µ).(1.543)H.
Kleinert, PATH INTEGRALS791.17 Classical and Quantum StatisticsThe average energy or internal energy is defined byE = Tr Ĥe−Ĥ/kB T.(1.544)F (T ).(1.546)Tr e−Ĥ/kB T .It may be obtained from the partition function Z(T ) by forming the temperaturederivative∂∂E = Z −1 kB T 2Z(T ) = kB T 2log Z(T ).(1.545)∂T∂TIn terms of the free energy (1.542), this becomes∂∂(−F (T )/T ) = 1 − TE=T∂T∂T2!For a grand-canonical ensemble we may introduce an average particle numberdefined by.(1.547)N = Tr N̂e−(Ĥ−µN̂ )/kB T Tr e−(Ĥ−µN̂ )/kB T .This can be derived from the grand-canonical partition function asN = ZG −1 (T, µ)kB T∂∂ZG (T, µ) = kB Tlog ZG (T, µ),∂µ∂µ(1.548)or, using the grand-canonical free energy, asN =−∂F (T, µ).∂µ G(1.549)The average energy in a grand-canonical system,E = Tr Ĥe−(Ĥ−µN̂ )/kB T.Tr e−(Ĥ−µN̂ )/kB T ,(1.550)can be obtained by forming, by analogy with (1.545) and (1.546), the derivativeE − µN = ZG −1 (T, µ)kB T 2=∂1−T∂T!∂Z (T, µ)∂T G(1.551)FG (T, µ).For a large number of particles, the density is a rapidly growing function ofenergy.
For a system of N free particles, for example, the number of states up toenergy E is given byN(E) =XpiΘ(E −NXp2i /2M),(1.552)i=1where each of the particle momenta pi is summed over all discrete momenta pm in(1.179) available to a single particle in a finite box of volume V = L3 . For a largeV , the sum can be converted into an integral24N(E) = VN"N ZYi=124NXd 3 pip2i /2M),3 Θ(E −(2πh̄)i=1#(1.553)Remember, however, the exception noted in the footnote to Eq. (1.184) for systems possessinga condensate.801 Fundamentalshwhich is simply V /(2πh̄)3√radius 2ME:iNtimes the volume Ω3N of a 3N-dimensional sphere ofN(E) ="V(2πh̄)3#N≡"V(2πh̄)3#NΩ3N(2πME)3N/2Γ3N2+1(1.554).Recall the well-known formula for the volume of a unit sphere in D dimensions:ΩD = π D/2 /Γ(D/2 + 1).(1.555)The surface is [see Eqs.
(8.116) and (8.117) for a derivation]SD = 2π D/2 /Γ(D/2).(1.556)Therefore, the density per energy ρ = ∂N /∂E is given by"Vρ(E) =(2πh̄)3#N(2πME)3N/2−1.2πMΓ( 32 N)(1.557)It grows with the very large power E 3N/2 in the energy. Nevertheless, the integral forthe partition function (1.578) is convergent, due to the overwhelming exponentialfalloff of the Boltzmann factor, e−E/kB T . As the two functions ρ(e) and e−e/kB Tare multiplied with each other, the result is a function which peaks very sharplyat the average energy E of the system.
The position of the peak depends on thetemperature T . For the free N particle system, for example,ρ(E)e−E/kB T ∼ e(3N /2−1) log E−E/kB T .(1.558)This function has a sharp peak atE(T ) = kB T3N3N− 1 ≈ kB T.22(1.559)The width of the peak is found by expanding (1.558) in δE = E − E(T ):)(E(T )13N3Nlog E(T ) −−(δE)2 + . . . .(1.560)exp22kB T2E (T ) 2√Thus, as soon as δE gets to be of the order of E(T )/ N , the exponential is reducedby a factor√ two with respect to E(T ) ≈ kB T 3N/2. The deviation is of a relativeorder 1/ N , i.e., the√peak is very sharp. With N being very large, the peak atE(T ) of width E(T )/ N can be idealized by a δ-function, and we may writeρ(E)e−E/kB T ≈ δ(E − E(T ))N(T )e−E(T )/kB T .(1.561)H. Kleinert, PATH INTEGRALS811.17 Classical and Quantum StatisticsThe quantity N(T ) measures the total number of states over which the system isdistributed at the temperature T .The entropy S(T ) is now defined in terms of N(T ) byN(T ) = eS(T )/kB .(1.562)Inserting this with (1.561) into (1.578), we see that in the limit of a large numberof particles N:Z(T ) = e−[E(T )−T S(T )]/kB T .(1.563)Using (1.542), the free energy can thus be expressed in the formF (T ) = E(T ) − T S(T ).(1.564)By comparison with (1.546) we see that the entropy may be obtained from the freeenergy directly as∂S(T ) = − F (T ).(1.565)∂TFor grand-canonical ensembles we may similarly considerZG (T, µ) =ZdE dn ρ(E, n)e−(E−µn)/kB T ,(1.566)whereρ(E, n)e−(E−µn)/kB T(1.567)is now strongly peaked at E = E(T, µ), n = N(T, µ) and can be written approximately asρ(E, n)e−(E−µn)/kB T ≈ δ (E − E(T, µ)) δ (n − N(T, µ))× eS(T,µ)/kB e−[E(T,µ)−µN (T,µ)]/kB T .(1.568)Inserting this back into (1.566) we find for large NZG (T, µ) = e−[E(T,µ)−µN (T,µ)−T S(T,µ)]/kB T .(1.569)For the grand-canonical free energy (1.543), this implies the relationFG (T, µ) = E(T, µ) − µN(T, µ) − T S(T, µ).(1.570)By comparison with (1.551) we see that the entropy can be calculated directly fromthe derivative of the grand-canonical free energyS(T, µ) = −∂F (T, µ).∂T G(1.571)The particle number is, of course, found from the derivative (1.549) with respect tothe chemical potential, as follows directly from the definition (1.566).821 FundamentalsThe canonical free energy and the entropy appearing in the above equationsdepend on the particle number N and the volume V of the system, i.e., they aremore explicitly written as F (T, N, V ) and S(T, N, V ), respectively.In the arguments of the grand-canonical quantities, the particle number N isreplaced by the chemical potential µ.Among the arguments of the grand-canonical energy FG (T, µ, V ), the volume Vis the only one which grows with the system.
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