44 Statistical Physics 44.1 Introduction; Boltzmann-Gibbs Probabilities Consider a quantum mechanical system (confined in a large volume), for which all energy values are discrete. Then ˆ Hψ j = E j ψ j , where ˆ H is the Hamilton operator and ψ l the complete set of orthogonal, nor- malized eigenfunctions of ˆ H. The observables of the system are represented by Hermitian 1 operators ˆ A, i.e., with real eigenvalues. For example, the spatial representation of the operator ˆp is given by the differential operator ˆp = i ∇ and the space operator ˆr by the corresponding multiplication operator. = h/(2π) is the reduced Planck’s constant, and i the square root of minus one (the imaginary unit). The eigenvalues of ˆ A are real, as well as the expectation values ψ j | ˆ Aψ j , which are the averages of the results of an extremely comprehensive series of measurements of ˆ A in the state ψ j . One can calculate these expectation val- ues, i.e., primary experimental quantities, theoretically via the scalar product given above, for example, in the one-particle spatial representation, without spin, as follows: ψ j | ˆ Aψ j = d 3 rψ ∗ j (r,t) ˆ A(ˆp, ˆr)ψ j (r,t) , where ψ j |ψ j = d 3 rψ ∗ j (r,t)ψ j (r,t) ≡ 1 . The thermal expectation value at a temperature T is then ˆ A T = j p j ·ψ j | ˆ Aψ j , (44.1) 1 more precisely: by self-adjoint operators, which are a) Hermitian and b) possess a complete system of eigenvectors. 344 44 Statistical Physics with the Boltzmann-Gibbs probabilities p j = e − E j k B T l e − E l k B T . Proof of the above expression will be deferred, since it relies on a so-called “ microcanonical ensemble ”, and the entropy of an ideal gas in this ensemble will first be calculated. In contrast to quantum mechanics, where probability amplitudes are added (i.e., ψ = c 1 ψ 1 + c 2 ψ 2 + ), the thermal average is incoherent. This follows explicitly from (44.1). This equation has the following interpretation: the sys- tem is with probability p j in the quantum mechanical pure state ψ j 2 ,and the related quantum mechanical expectation values for j =1, 2, ,which are bilinear expressions in ψ j , are then added like intensities as in incoher- ent optics, and not like amplitudes as in quantum mechanics. It is therefore important to note that quantum mechanical coherence is destroyed by ther- malization. This limits the possibilities of “quantum computing” discussed in Part III (Quantum Mechanics). The sum in the denominator of (44.1), Z(T ):= l e − E l k B T , is the partition function. This is an important function, as shown in the following section. 44.2 The Harmonic Oscillator and Planck’s Formula The partition function Z(T ) can in fact be used to calculate “almost any- thing ”! Firstly, we shall consider a quantum mechanical harmonic oscillator. The Hamilton operator is ˆ H = ˆp 2 2m + mω 2 0 2 ˆx 2 with energy eigenvalues E n = n + 1 2 ·ω 0 , with n =0, 1, 2, . Therefore Z(T )= ∞ n 0 e −β·(n+ 1 2 )·ω 0 , with β = 1 k B T . 2 or the corresponding equivalence class obtained by multiplication with a complex number 44.2 The Harmonic Oscillator and Planck’s Formula 345 Thus Z(T )=e βω 0 2 · ˜ Z(β) , with ˜ Z(β):= ∞ n=0 e −n·βω 0 . From Z(T ) we obtain, for example: U(T )= ˆ H T = ∞ n=0 p n · E n = ∞ n=0 n + 1 2 ω 0 ·p n = ω 0 · ⎛ ⎜ ⎜ ⎝ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∞ n=0 ne −(n+ 1 2 )βω 0 ∞ n=0 e −(n+ 1 2 )βω 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + 1 2 ⎞ ⎟ ⎟ ⎠ = ω 0 × ⎛ ⎜ ⎜ ⎝ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∞ n=0 ne −nβω 0 ∞ n=0 e −nβω 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + 1 2 ⎞ ⎟ ⎟ ⎠ = − d dβ ln ˜ Z(β)+ ω 0 2 . But ˜ Z(β), which is required at this point, is very easy to calculate as it is an infinite geometric series: ˜ Z(β)= ∞ n=0 e −nβω 0 ≡ 1 1 −e −βω 0 . Finally we obtain U(T )=ω 0 · e −βω 0 1 −e −βω 0 + 1 2 = ω 0 · 1 e βω 0 − 1 + 1 2 = ˜ U(T )+ ω 0 2 , with ˜ U(T )= ω 0 e βω 0 − 1 . This expression corresponds to the Planck radiation formula mentioned pre- viously. As we have already outlined in Part III, Planck (1900) proceeded from the opposite point of view. Experiments had shown that at high temperatures ˜ U = k B T appeared to be correct (Rayleigh-Jeans) while at low temperatures ˜ U(T )=ω 0 e −βω 0 (Wien). Planck firstly showed that the expression ˜ U(T )= ω 0 e βω 0 − 1 346 44 Statistical Physics (or more precisely: the related expression for entropy) not only interpo- lated between those limits, but also reproduced all relevant experiments in- between. He then hit upon energy quantization and the sequence E n = nω 0 , at that time without the zero-point energy ω 0 2 . Indeed, the expression ˜ U(T )= ω 0 e βω 0 − 1 describes the experimentally observed behavior at both high and low tem- peratures, for example, for k B T ω 0 (high T )wehave: e ω 0 k B T − 1 ≈ ω 0 k B T . The expression ˜ U(T )= ω 0 e βω 0 − 1 is valid for a single harmonic oscillator of frequency ν = ω 0 2π . One can regard ω 0 as the excitation energy of an individual quantum of this electrodynamic oscillation mode (→ photons). In solids other vibrational quanta of this kind exist (phonons, magnons, plasmons etc., see below), which are regarded as “ quasi-particles ”. They have zero chemical potential μ,such that the factor n T := e βω 0 − 1 −1 can be interpreted as the thermal expectation value for the number of quasi- particles n.Thus U(T )=ω 0 · n T + 1 2 . (Cf. one of the last sections in Part II, Sect. 21.1, dealing with dispersion, or Sect. 53.5 below.) To summarize, plane electromagnetic waves of wavenumber k =2π/λ travelling with the speed of light c in a cavity of volume V , together with the relation λ = c/ν =2πc/ω 0 between wavelength λ and frequency ν (or angular frequency ω 0 )canbein- terpreted as the vibrational modes of a radiation field. Formally this field has 44.2 The Harmonic Oscillator and Planck’s Formula 347 the character of an ensemble of oscillators, the particles (or quasi-particles) of the field, called photons (or the like). Therefore from ˜ U(T )oneobtains Planck’s radiation formula by multiplying the result for a single oscillator with V · 2 ·d 3 k/(2π) 3 = V ·8πν 2 dν/c 3 . (The factor 2 arises because every electromagnetic wave, since it is transverse, must have two linearly independent polarization directions.) If there is a small hole (or aperture)ofareaΔ 2 S in the outer boundary of the cavity, it can be shown that in a time Δt the amount of energy that escapes from the cavity as radiation is given by ΔI E := ∞ 0 c 4 ·u T (ω 0 )dω 0 · Δ 2 SΔt . One can thus define a related spectral energy flux density j E (ω 0 )ofthecavity radiation j E (ω 0 )= c 4 ·u T (ω 0 ) . (44.2) (The density u T (ω 0 ) corresponds to the quantity ˜ U(T ).) This expression is utilized in bolometry, where one tries to determine the energy flux and tem- perature from a hole in the cavity wall 3 . 3 A corresponding exercise can be found on the internet, see [2], winter 1997, file 8. 45 The Transition to Classical Statistical Physics 45.1 The Integral over Phase Space; Identical Particles in Classical Statistical Physics The transition to classical statistical physics is obtained by replacing sums of the form l p l · g(E l ) by integrals over the phase space of the system. For f degrees of freedom we obtain: p 1 x 1 p f x f dp 1 dx 1 · ·dp f dx f h f · g(H(p 1 , ,p f ,x 1 , ,x f )) , where h is Planck’s constant. Here the quasi-classical dimensionless measure of integration dp 1 dx 1 · ·dp f dx f h f is, so to speak, quantized in units of h f ; one can show that this result is invariant for Hamilton motion and corresponds exactly to the factor V · d 3 k/(2π) 3 (see below) in the Planck formula. It is also obtained with f =3andthede Broglie relation p = k , as well as V = Δ 3 x. With regard to phase-space quantization it is appropriate to mention here Heisenberg’s uncertainty principle in the form ΔxΔp x ∼ h. The factor V · d 3 k/(2π) 3 indeed follows by taking a wave-field approach `aladeBroglie: ψ(x) ∝ e ik·r , 350 45 The Transition to Classical Statistical Physics and replacing the system volume by a box of equal size with periodic boundary conditions, which practically does not alter the volume properties, but makes counting the eigenmodes easier. For a one-dimensional system of size L, due to ψ(x + L) ≡ ψ(x) , we have: k ≡ 2πn L , where n is integer. Thus ∞ −∞ dn · = L 2π · ∞ −∞ dk · , and in three dimensions d 3 n · = V (2π) 3 · d 3 k · . To summarize, the transition from a statistical treatment of quantum sys- tems to classical statistical physics corresponds completely with the wave pic- ture of matter of Louis de Broglie, which was indeed fundamental for the introduction of Schr¨odinger’s equation (→ Quantum Mechanics).Thefac- tors h −f cancel each other out when calculating expectation values, ˆ A(p, x) T = dpdx ˆ A(ˆp, ˆx) dpdx ; but for calculating the entropy (see below) they give non-trivial contributions. There is a further (perhaps even more important) “ legacy ” from quan- tum mechanics: the effective indistinguishability of identical particles.ForN identical particles, whether fermions or bosons, one must introduce a permu- tation factor 1/N ! in front of the integral in phase space (and of course put f =3N). Sackur and Tetrode (1913) pointed out long before quantum me- chanics had been introduced that the basic property of additivity of entropy, which had been required since Boltzmann, is only obtained by including this permutation factor, even for the case of an ideal gas (see below). The fact that even from the viewpoint of classical physics the molecules of an ideal gas are indistinguishable is an astonishing suggestion, because one could indeed have the idea of painting them in N different colors to make them distinguishable. This could involve a negligible amount of work, but it would significantly change the entropy of the system (see again later). 45.2 The Rotational Energy of a Diatomic Molecule The rotational behavior of diatomic molecules will now be discussed in more detail. This section refers in general to elongated molecules, i.e. where the longitudinal moment of inertia of the molecule Θ || is negligible compared 45.2 The Rotational Energy of a Diatomic Molecule 351 to the transverse moment of inertia Θ ⊥ . In addition the ellipsoid describing the anisotropy of the moment of inertia is assumed here to be a rotational ellipsoid (→ Part I). The rotational energy of such a molecule, on rotation about an axis per- pendicular to the molecular axis, is found from the Hamiltonian ˆ H = ˆ L 2 2Θ ⊥ . Due to quantization of the angular momentum the partition function is given by Z(T )= ∞ l=0 (2l +1)· e −β 2 l·(l+1) 2Θ ⊥ . The factor (2l + 1) describes the degeneracy of the angular momentum, i.e. the states Y l,m l (ϑ, ϕ) , for m l = −l, −l +1, ,+l, have the same eigenvalues, 2 l ·(l +1), of the operator ˆ L 2 . At very low temperatures, k B T 2 · 2Θ ⊥ , rotation is quenched due to the finite energy gap Δ := 2 Θ ⊥ betweenthegroundstate(l = 0) and the (triplet) first excited state (l =1). The characteristic freezing temperature Δ/k B lies typically at approximately 50 K. At high temperatures, e.g. room temperature, rotations with l 1are almost equally strongly excited, so that we can approximate the sum Z(T )= ∞ l=0 (2l +1)· e −β 2 l·(l+1) 2Θ ⊥ by the integral ∞ l=0 dl · 2l ·e −β ˆ L 2 2Θ ⊥ . Substituting (still for l 1) l → L , it follows that 2l · dl ≈ 1 π d 2 L 2 . 352 45 The Transition to Classical Statistical Physics We thus arrive at exactly the classical result with two additional degrees of freedom per molecule and an energy k B T 2 per degree of freedom.Thislast result is called the law of equipartition of energy and is valid for canonical variables p i and q i which occur as bilinear terms in the classical Hamilton function. The vibrations of diatomic molecules also involve two additional degrees of freedom per molecule (kinetic plus potential energy), but at room temper- ature they are generally still frozen-in, in contrast to the rotational degrees of freedom. The low-temperature behavior of diatomic molecules is thus determined by quantum mechanics. For example, U(T )=[E 0 g 0 e −βE 0 + E 1 g 1 e −βE 1 + ]/[g 0 e −βE 0 + g 1 e −βE 1 + ] , with degeneracy factors g k , k =0, 1, 2 ,wherewewriteE 0 = 0. The high- temperature behavior follows classical statistical physics, for example, U(T )=f · k B T 2 . The transition region satisfies the relation k B T ≈ (E 1 − E 0 ); the transition itself can be calculated numerically using the Euler-MacLaurin summation formula. 1 (Remarkably, it is non-monotonic.) With the above integral expression for classical statistical physics we also automatically obtain as special cases: a) the Maxwell-Boltzmann velocity distribution: F (v)d 3 v ∝ e −β mv 2 2 4πv 2 dv. (The proportionality constant is obtained from the identity R 3 d 3 x e − x 2 2σ 2 (2πσ 2 ) 3 2 =1.) b) the barometric pressure equation: n(z) n(0) = p(z) p(0) =exp(−βmgz) . Alternative derivations for both the Maxwell-Boltzmann velocity distribution and the barometric formula can be found in many textbooks, which provide plausible verification of the more general Boltzmann-Gibbs distribution. This means that one can arrive at these results by avoiding the transition from a microcanonical to a canonical or grand canonical ensemble, for example, if one wishes to introduce the Boltzmann-Gibbs distribution into the school curriculum. 1 More details can be found in the “G¨oschen” booklet on Theoretical Physics by D¨oring, [39], Band V, Paragraph 16 (in German). 46 Advanced Discussion of the Second Law 46.1 Free Energy We next define the quantity Helmholtz Free Energy F (T,V,m H ,N):=U(T,V,m H ,N) − T · S(T,V,m H ,N) . The meaning of this thermodynamic variable is “available energy”, i.e., taking into account the heat loss, given by the entropy S multiplied by the absolute temperature T, which is subtracted from the internal energy. Indeed one arrives at this interpretation by forming the differential of F andthenusing both the first and second laws: dF =dU − T · dS −S ·dT = δA + δQ − T ·dS −S ·dT ≤ δA − S ·dT. This inequality, dF ≤ δA −S ·dT, is actually a way of stating the first and second laws simultaneously. In par- ticular, under isothermal conditions we have dF ≤ δA T . For a reversible process the equality holds. Then dF = −pdV + Hdm H + μdN − SdT, i.e., ∂F ∂V = −p ; ∂F ∂m H = H ; ∂F ∂N = μ and ∂F ∂T = −S. By equating the mixed second derivatives ∂ 2 F ∂x 1 ∂x 2 = ∂ 2 F ∂x 2 ∂x 1 one obtains further Maxwell relations: ∂S ∂V = ∂p ∂T , ∂p ∂m H = − ∂H ∂V and ∂μ ∂T = − ∂S ∂N . . Oscillator and Planck’s Formula The partition function Z(T ) can in fact be used to calculate “almost any- thing ”! Firstly, we shall consider a quantum mechanical harmonic oscillator. The Hamilton. (or angular frequency ω 0 )canbein- terpreted as the vibrational modes of a radiation field. Formally this field has 44.2 The Harmonic Oscillator and Planck’s Formula 347 the character of an ensemble. numerically using the Euler-MacLaurin summation formula. 1 (Remarkably, it is non-monotonic.) With the above integral expression for classical statistical physics we also automatically obtain as