Basic Theoretical Physics A Concise Overview

44 Statistical Physics44.1 Introduction; Boltzmann-Gibbs ProbabilitiesConsider a quantum mechanical system (confined in a large volume), forwhich all energy values are discrete. Then Hˆ ψj = Ejψj ,where Hˆ is the Hamilton operator and ψl the complete set of orthogonal, nor-malized eigenfunctions of Hˆ . The observables of the system are represented byHermitian1 operators Aˆ, i.e., with real eigenvalues. For example, the spatialrepresentation of the operator pˆ is given by the differential operator pˆ = ∇ iand the space operator rˆ 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 Aˆ are real, as well as the expectationvalues ψj |Aˆ ψj ,which are the averages of the results of an extremely comprehensive series ofmeasurements of Aˆ in the state ψj. One can calculate these expectation val-ues, i.e., primary experimental quantities, theoretically via the scalar productgiven above, for example, in the one-particle spatial representation, withoutspin, as follows: ψj|Aˆ ψj = d3rψj∗(r, t)Aˆ(pˆ, rˆ)ψj (r, t) , where ψj|ψj = d3rψj∗(r, t)ψj (r, t) ≡ 1 .The thermal expectation value at a temperature T is then Aˆ = pj · ψj|Aˆψj , (44.1) T j1 more precisely: by self-adjoint operators, which are a) Hermitian and b) possess a complete system of eigenvectors.

344 44 Statistical Physicswith the Boltzmann-Gibbs probabilities pj = e− Ej . kB T e− El kB T lProof 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 ensemblewill first be calculated. In contrast to quantum mechanics, where probability amplitudes are added(i.e., ψ = c1ψ1 + c2ψ2 + . . .), the thermal average is incoherent. This followsexplicitly from (44.1). This equation has the following interpretation: the sys-tem is with probability pj in the quantum mechanical pure state ψj2, andthe related quantum mechanical expectation values for j = 1, 2, . . ., whichare bilinear expressions in ψj , are then added like intensities as in incoher-ent optics, and not like amplitudes as in quantum mechanics. It is thereforeimportant to note that quantum mechanical coherence is destroyed by ther-malization. This limits the possibilities of “quantum computing” discussed inPart III (Quantum Mechanics). The sum in the denominator of (44.1), El kB T Z(T ) := e ,− lis the partition function. This is an important function, as shown in thefollowing section.44.2 The Harmonic Oscillator and Planck’s FormulaThe 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 Hˆ = pˆ2 + mω02 xˆ2 2m 2with energy eigenvalues En = 1 · ω0 , with n = 0, 1, 2, . . . . n+ 2Therefore ∞ Z(T ) = e−β·(n+ 1 )· ω0 , with 1 2 β= . n0 kB T2 or the corresponding equivalence class obtained by multiplication with a complex number

44.2 The Harmonic Oscillator and Planck’s Formula 345Thus ∞ Z(T ) = β ω0 · Z˜(β) , with Z˜(β) := e−n·β ω0 . 2 e n=0From Z(T ) we obtain, for example: ∞∞ 1 ω0 · pn n+U (T ) = Hˆ T = pn · En = 2 n=0 n=0 ⎛⎧ ⎫ ⎞ ω0 ⎜⎜⎝⎪⎪⎪⎪⎨⎩ ∞ ne−(n+ 1 )β ω0 ⎪⎪⎬ ⎠⎟⎟ 2 ω0 ⎭⎪⎪ n=0 = · ∞ + 1 2 e−(n+ 1 )β 2 = ω0 × ⎜⎛⎝⎜⎧⎪⎪⎩⎨⎪⎪nn=∞=∞00 ne−nβ ω0 ⎫ + 1 ⎞ e−nβ ω0 ⎪⎪⎬ 2 ⎠⎟⎟ ⎭⎪⎪ n=0 = − d ln Z˜(β) + ω0 . dβ 2But Z˜(β), which is required at this point, is very easy to calculate as it is aninfinite geometric series: ∞ 1 1 − e−β Z˜(β) = e−nβ ω0 ≡ . ω0 n=0Finally we obtain U (T ) = ω0 · e−β ω0 1 = ω0 · 11 1 − e−β ω0 + 2 eβ ω0 − 1 + 2 = U˜ (T ) + ω0 , with U˜ (T ) = eβ ω0 . 2 ω0 − 1This expression corresponds to the Planck radiation formula mentioned pre-viously. As we have already outlined in Part III, Planck (1900) proceeded from theopposite point of view. Experiments had shown that at high temperaturesU˜ = kBT appeared to be correct (Rayleigh-Jeans) while at low temperatures U˜ (T ) = ω0e−β ω0(Wien). Planck firstly showed that the expression U˜ (T ) = eβ ω0 ω0 − 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 En = n ω0 ,at that time without the zero-point energy ω0 . 2Indeed, the expression U˜ (T ) = eβ ω0 ω0 − 1describes the experimentally observed behavior at both high and low tem-peratures, for example, for kBT ω0 (high T ) we have: ω0 ω0 . kB T e kB T − 1 ≈The expression U˜ (T ) = eβ ω0 ω0 − 1is valid for a single harmonic oscillator of frequency ν = ω0 . 2πOne can regard ω0 as the excitation energy of an individual quantum ofthis electrodynamic oscillation mode (→ photons). In solids other vibrationalquanta of this kind exist (phonons, magnons, plasmons etc., see below), whichare regarded as “ quasi-particles ”. They have zero chemical potential μ, suchthat the factor n T := eβ ω0 − 1 −1can be interpreted as the thermal expectation value for the number of quasi-particles n. Thus U (T ) = ω0 · 1 . nT+2(Cf. one of the last sections in Part II, Sect. 21.1, dealing with dispersion, orSect. 53.5 below.) To summarize, plane electromagnetic waves of wavenumber k = 2π/λtravelling with the speed of light c in a cavity of volume V , together with therelation λ = c/ν = 2πc/ω0between wavelength λ and frequency ν (or angular frequency ω0) can be in-terpreted as the vibrational modes of a radiation field. Formally this field has