Dilute Polyatomic Gases, Vibrations of a Solid, Black-Body Radiation

VI. Quantum Statistical Mechanics There are limitations to the applicability of classical statistical mechanics. The need to include quantum mechanical effects becomes specially apparent at low temperatures. In this section we shall first demonstrate the failure of the classical results in the contexts of heat capacities of molecular gases and solids, and the ultra-violet catastrophe in black body radiation. We shall then reformulate statistical mechanics using quantum concepts. VI.A Dilute Polyatomic Gases Consider a dilute gas of polyatomic molecules. The Hamiltonian for each molecule of n atoms is H1 = n X p� 2i + V(�q1 , . . . , �qn ), 2m (VI.1) i=1 where the potential energy V, contains all the information on molecular bonds. For sim­ plicity, we have assumed that all atoms in the molecule have the same mass. If the masses are different, the Hamiltonian can be brought into the above form by rescaling the coordi­ p p nates �qi by mi /m (and the momenta by m/mi ), where mi is the mass of the ith atom. Ignoring the interactions between molecules, the partition function of a dilute gas is ZN 1 Z(N ) = 1 = N! N! (Z n " #)N n X Y d3 p�i d3 �qi � 2i p exp −β − βV(q�1 , . . . , �qn ) . 3 h 2 m i=1 i=1 (VI.2) The chemical bonds that keep the molecule together are usually quite strong (ener­ gies of the order of electron volts). At typical accessible temperatures, which are much smaller than the corresponding dissociation temperatures (≈ 104 ◦ K), the molecule has a well defined shape and only undergoes small deformations. The contribution of these deformations to the one particle partition function Z1 , can be computed as follows: (a) The first step is to find the equilibrium positions, (�q ∗1 , . . . , �q n∗ ), by minimizing the potential V. (b) The energy cost of small deformations about equilibrium is then obtained by setting �qi = �q ∗i + �ui , and making an expansion in powers of �u, n 3 ∂ 2V 1 X X V =V + ui,α uj,β + O(u3 ). 2 i,j=1 ∂qi,α ∂qj,β ∗ α,β=1 118 (VI.3) (Here i, j = 1, · · · , n, identify the atoms, and α, β = 1, 2, 3 label a particular com­ ponent.) Since the expansion is around a stable equilibrium configuration, the first derivatives are absent in eq.(VI.3), and the matrix of second derivatives is positive definite, i.e. it has only non-negative eigenvalues. (c) The normal modes of the molecule are obtained by diagonalizing the 3n × 3n matrix ∂ 2 V/∂qi,α ∂qj,β . The resulting 3n eigenvalues Ks , indicate the stiffness of each mode. We can change variables from the original deformations {�ui }, to the amplitudes {ũs }, of the eigenmodes. The corresponding conjugate momenta are p̃s = mũ˙s . Since the transformation from {�ui } to {ũs } is unitary (preserving the length of a vector), P P �i 2 = s p̃2s , and the quadratic part of the resulting deformation Hamiltonian is ip  3n  X 1 2 Ks 2 H1 = V + p̃ + ũ . 2m s 2 s s=1 ∗ (VI.4) (Such transformations are also canonical, preserving the measure of integration in Q Q phase space, i,α dui,α dpi,α = s dũs dp̃s .) The average energy of each molecule is the expectation value of the above Hamiltonian. Since each quadratic degree of freedom classically contributes a factor of kB T /2 to the energy, 3n + m (VI.5) kB T. 2 Only modes with a finite stiffness can store potential energy, and m is defined as the �H1 � = V ∗ + number such modes with non-zero Ks . The following symmetries of the potential force some eigenvalues to zero: Since V(q�1 + �c, · · · , �qn + �c ) = V(q�1 , · · · , �qn ), no energy is P � ) = V(Q � + �c ), and the � = qα /n, i.e. V(Q stored the center of mass coordinate Q α� (a) Translation symmetry: corresponding three values of Ktrans are zero. (b) Rotation symmetry: There is also no potential energy associated with rotations of the molecule, and Krot = 0 for the corresponding stiffnesses. The number of rota­ tional modes, 0 ≤ r ≤ 3, depends on the shape of the molecule; for example, a rod shaped molecule has r = 2, as a rotation parallel to its axis does not result in a new configuration. The remaining m = 3n − 3 − r eigenvectors of the matrix have non-zero stiffness, and correspond to the vibrational normal modes. The energy per molecule, from eq.(VI.5), is thus �H1 � = 6n − 3 − r kB T. 2 119 (VI.6) The corresponding heat capacities, CV = (6n − 3 − r) kB , 2 and CP = CV + kB = (6n − 1 − r) kB , 2 (VI.7) are temperature independent. The ratio γ = CP /CV is easily measured in adiabatic processes. Values of γ, expected on the basis of the above argument, are listed below for a number of different molecules. Monatomic He n=1 r=0 γ = 5/3 Diatomic O2 or CO n=2 r=2 γ = 9/7 n=3 r=2 γ = 15/13 n=3 r=3 γ = 14/12 = 7/6 n=4 r=3 γ = 20/18 = 10/9 Linear triatomic O–C–O Planar triatomic H /O \ H Tetra-atomic NH3 Measurements of the heat capacity of dilute gases do not agree with the above pre­ dictions. For example, the value CV /kB = 7/2, for a diatomic gas such as oxygen, is only observed at temperatures higher than a few thousand degrees Kelvin. At room tempera­ tures, a lower value of 5/2 is observed, while at even lower temperatures of around 100 K, it is further reduced to 3/2. The low temperature value is similar to that of a monatomic gas, and suggests that no energy is stored in the rotational and vibrational degrees of freedom. These observations can be explained if the allowed energy levels are quantized. • Vibrational modes: A diatomic molecule has one vibrational mode with stiffness K ≡ mω 2 , where ω is the frequency of oscillations. The classical partition function for this mode is c Zvib    2 dp dq p mω 2 q 2 = exp −β + h 2m 2 s   1 2πm 2π 2π kB T = = = , 2 hβω ¯ hω h β βmω Z (VI.8) where h̄ = h/2π. The corresponding energy stored in this mode, �Hvib �c = − ∂ ln Z ∂ ln(β¯hω) 1 = = = kB T, ∂β ∂β β (VI.9) comes from kB T /2 per kinetic and potential degrees of freedom. In quantum mechanics, the allowed values of energy are quantized such that Hqvib  1 = hω ¯ n+ 2 120  , (VI.10) with n = 0, 1, 2, · · ·. Assuming that the probability of each discrete level is proportional to its Boltzmann weight (as will be justified later on), there is a normalization factor q Zvib = ∞ X e−β¯hω(n+1/2) = n=0 e−β¯hω/2 . 1 − e−β¯hω (VI.11) The high temperature limit, q lim Zvib = β→0 1 kB T = , β¯hω ¯hω coincides with eq.(VI.8)(due in part to the choice of h as the measure of classical phase space). The expectation value of vibrational energy is q =− Evib hω hω e−β¯hω ∂ ln Z ¯ ∂ ¯ + ln(1 − e−β¯hω ) = + ¯hω . = ∂β 2 ∂β 2 1 − e−β¯hω (VI.12) The first term is the energy cost of quantum fluctuations that are present even in the zero temperature ground state. The second term describes the additional energy due to thermal fluctuations. The resulting heat capacity, q Cvib q dEvib = = kB dT  ¯hω kB T 2 e−β¯hω , (1 − e−β¯hω )2 (VI.13) achieves the classical value of kB only at temperatures T ≫ θvib , where θvib = h̄ω/kB is a characteristic temperature associated with the quanta of vibrational energy. For T ≪ θvib , q goes to zero as exp(−θvib /T ). Typical values of θvib are in the range of 103 to 104 Cvib degrees Kelvin, explaining why the classical value of heat capacity is observed only at higher temperatures. • Rotational modes: To account for the low temperature anomaly in the heat capacity of diatomic molecules, we have to study the quantization of the rotational degrees of freedom. Classically, the orientation of a diatomic molecule is specified by two angles θ and φ, and its Lagrangian (equal to the kinetic energy) is L=  I 2 θ̇ + sin2 θ φ̇2 , 2 (VI.14) where I is the moment of inertia. In terms of the conjugate momenta, pθ = ∂L = Iθ̇ , dθ̇ pφ = 121 ∂L ˙ = I sin2 θ φ, dφ̇ (VI.15) the Hamiltonian for rotations is Hrot 1 = 2I p2φ p2θ + sin2 θ ! ≡ �2 L , 2I (VI.16) � is the angular momentum. From the classical partition function, where L c Zrot " Z π Z 2π Z ∞ 1 β = 2 dθ dφ dpθ dpφ exp − h 0 2I 0 −∞    2πI 4π 2IkB T = = , 2 β h h̄2 p2θ + p2φ sin2 θ !# (VI.17) the stored energy is ∂ ln Z ∂ �Erot � = − = ln ∂β ∂β c  βh̄2 2I  = kB T, (VI.18) as expected for two degrees of freedom. In quantum mechanics, the allowed values of � 2 = h̄2 ℓ(ℓ + 1) with ℓ = 0, 1, 2, · · ·, and each state angular momentum are quantized to L has a degeneracy of 2ℓ + 1 (along a selected direction, Lz = −ℓ, · · · , +ℓ). A partition function is now obtained for these levels as q Zrot = ∞ X ℓ=0     ∞ X βh̄2 ℓ(ℓ + 1) θrot ℓ(ℓ + 1) exp − (2ℓ + 1) = exp − (2ℓ + 1), 2I T (VI.19) ℓ=0 where θrot = h̄2 /(2IkB ) is a characteristic temperature associated with quanta of rotational energy. While the sum can not be analytically evaluated in general, we can study its high and low temperature limits: (a) For T ≫ θrot , the terms in eq.(VI.19) vary slowly, and the sum can be replaced by the integral lim T →∞ q Zrot ∞  θrot x(x + 1) = dx(2x + 1) exp − T Z0 ∞ T c = dy e−θrot y/T = = Zrot , θ rot 0 Z  (VI.20) i.e. the classical result of eq.(VI.17) is recovered. (b) For T ≪ θrot , the first few terms dominate the sum, and q = 1 + 3e−2θrot /T + O(e−6θrot /T ), lim Zrot T →∞ 122 (VI.21) leading to an energy q =− Erot h i ∂ ln Z ∂ ≈− ln 1 + 3e−2θrot /T ≈ 6kB θrot e−2θrot /T . ∂β ∂β (VI.22) The resulting heat capacity vanishes at low temperatures as Crot q dErot = = 3kB dT  2θrot T 2 e−2θrot /T + · · · . (VI.23) Typical values of θrot are between 1 and 10 ◦ K, explaining the lower temperature shoulder in the heat capacity measurements. At very low temperatures, the only contributions come from the kinetic energy of the center of mass, and the molecule behaves as a monatomic particle. (The heat capacity vanishes at even lower temperatures due to quantum statistics, as will be discussed in the context of identical particles.) VI.B Vibrations of a Solid Attractive interactions between particles first lead to condensation from a gas to liquid at low temperatures, and finally cause freezing into a solid state at even lower temperatures. For the purpose of discussing its thermodynamics, the solid can be regarded as a very large molecule subject to a Hamiltonian similar to eq.(VI.1), with n = N ≫ 1 atoms. We can then proceed with the steps outlined in the previous section. (a) The classical ground state configuration of the solid is obtained by minimizing the po­ tential V. In almost all cases, the minimum energy corresponds to a periodic arrangement of atoms forming a lattice. In terms of the three basis vectors, â, b̂, and ĉ, the locations of atoms in a simple crystal are given by h i �q (ℓ, m, n) = ℓâ + mb̂ + nĉ ≡ �r, ∗ (VI.24) where {ℓ, m, n} is a triplet of integers. (b) At finite temperatures, the atoms may undergo small deformations �q�r = �r + �u(�r ), (VI.25) with a cost in potential energy of V = V∗ + 1 X ∂ 2V uα (�r ) uβ (�r ′ ) + O(u3 ). ′ ∂q ∂q 2 � r,α � r ,β ′ � r,� r α,β 123 (VI.26) (c) Finding the normal modes of a crystal is considerably simplified by its translational symmetry. In particular, the matrix of second derivatives only depends on the relative separation of two points, ∂ 2V = Kαβ (�r − �r ′ ). ′ ∂q�r,α ∂q�r ,β (VI.27) It is always possible to take advantage of such a symmetry to at least partially diagonalize the matrix by using a Fourier basis, uα (�r ) = ′ � X eik·�r √ ũα (�k ). N � (VI.28) k The sum is restricted to wavevectors �k inside a Brillouin zone. For example, in a cu­ bic lattice of spacing a, each component of �k is restricted to the interval [−π/a, π/a]. This is because wavevectors outside this interval carry no additional information as (kx + 2πm/a)(na) = kx (na) + 2mnπ, and any phase that is a multiple of 2π does not effect the sum in eq.(VI.28). In terms of the Fourier modes, the potential energy of defor­ mations is V = V∗ + 1 2N X � (� r,� r ′ ),(� k,� k ′) α,β � ′ ·� r′ Kαβ (�r − �r ′ )eik·�r ũα (�k )eik ũβ (�k ′ ). (VI.29) We can change variables to relative and center of mass coordinates, by setting ′ � = �r + �r , R 2 ρ � = �r − �r ′ , and � + ρ� , �r = R 2 � − ρ� . and �r ′ = R 2 Eq.(VI.29) now simplifies to    X X X 1 � �′ � � �′  V = V∗ + � )ei(k−k )·�ρ/2 ũα (�k )ũβ (�k ′ ) . Kαβ (ρ ei(k+k )·R   2N ′ � k,� k α,β � R (VI.30) ρ � As the sum in the first brackets is N δ�k+�k′ ,0 ,   X X 1 �  � )eik·ρ�  ũα (�k )ũβ (−�k ) V = V∗ + Kαβ (ρ 2 � k,α,β ρ � 1 X =V + K̃αβ (�k )ũα (�k )ũβ (�k )∗ , 2 ∗ � k,α,β 124 (VI.31) P where K̃αβ (�k ) = ρ� Kαβ (ρ � ) exp(i�k · ρ� ), and ũβ (�k )∗ = ũβ (−�k ) is the complex conjugate of ũβ (�k ). The different Fourier modes are thus decoupled at the quadratic order, and the task of diagonalizing the 3N × 3N matrix of second derivatives is reduced to diagonalizing the 3 × 3 matrix K̃αβ (�k ) separately for each �k. The from of K̃αβ is further restricted by the point group symmetries of the crystal. The discussion of such constraints is beyond the intent of this section and for simplicity we shall assume that K̃αβ (�k ) = δα,β K̃(�k ), is already diagonal. (For an isotropic material, this implies a specific relation between bulk and shear moduli.) The kinetic energy of deformations is N X m i=1 2 �q˙ 2i = Xm � k,α 2 u̇ ˜α (�k )u̇ ˜α (�k )∗ = X 1 pα (�k )∗ , p˜α (�k )˜ 2m (VI.32) � k,α where p̃α (�k ) = ∂L = mũ˙α (�k ), ∂ũ˙α (�k ) is the momentum conjugate to ũα (�k ). The resulting deformation Hamiltonian, H = V∗ + X � k,α " 1 p̃α (�k ) 2m 2 K̃(�k ) + ũα (�k ) 2 2 # , describes 3N independent harmonic oscillators of frequencies ωα (�k ) = (VI.33) q K̃(�k )/m. In a classical treatment, each harmonic oscillator of non-zero stiffness contributes kB T to the internal energy of the solid. At most, 6 of the 3N oscillators are expected to have zero stiffness (corresponding to uniform translations and rotations of the crystal). Thus, up to non-extensive corrections of order 1/N , the classical internal energy associated with Hamiltonian (VI.33) is 3N kB T , leading to a temperature independent heat capacity of 3kB per atom. In fact, the measured heat capacity vanishes at low temperatures. We can again relate this observation to the quantization of the energy levels of each oscillator, as discussed in the previous section. Quantizing each harmonic mode separately, gives the allowed values of the Hamiltonian as q ∗ H =V + X � k,α   1 h̄ωα (�k ) n�k,α + , 2 125 (VI.34) where the set of integers {n�k,α } describes the quantum micro-state of the oscillators. Since the oscillators are independent, their partition function, q Z = X −β H q e −βE0 =e YX −β¯ hωα (� k )n~k,α e −βE0 =e � k,α n~k,α {n~k,α } Y � k,α 1 1 − e−β¯hωα (�k )  , (VI.35) is the product of single oscillator partition functions such as eq.(VI.11). (E0 includes the ground state energies of all oscillators in addition to V ∗ .) The internal energy is E(T ) = �Hq � = E0 + X � k,α D E ¯hωα (�k ) nα (�k ) , (VI.36) where the average occupation numbers are given by D   E P∞ ne−β¯hωα (�k )n ∂ 1 n=0 � nα (k ) = P∞ =− ln −βhω ¯ α (� k )n �k )) ∂(βh̄ω e ( 1 − e−β¯hωα (�k ) α n=0 � = e−β¯hωα (k ) ¯ α (� k) 1 − e−βhω = 1 eβ¯hωα (�k ) − 1 (VI.37) . As a first attempt at including quantum mechanical effects, we may adopt the Einstein model in which all the oscillators are assumed to have the same frequency ωE . This model corresponds to atoms that are pinned to their ideal location by springs of stiffness 2 . The resulting internal energy, K̄ = ∂ 2 V/∂q 2 = mωE E = E0 + 3N h̄ωE e−β¯hωE , 1 − e−β¯hωE and heat capacity, dE C= = 3N kB dT  TE T 2 e−TE /T 1 − e−TE /T (VI.38)
2 , (VI.39) is simply proportional to that of a single oscillator (eqs.(VI.12) and (VI.13)). In particular, there is an exponential decay of the heat capacity to zero with a characteristic temperature ¯ E /kB . However, the experimentally measured heat capacity decays to zero much TE = hω more slowly, as T 3 . The discrepancy is resolved through the Debye model, which emphasizes that at low temperatures the main contribution to heat capacity is from the oscillators of lowest fre­ quency that are the most easily excited. The lowest energy modes in turn correspond to smallest wavevectors k = |�k|, or longest wavelengths λ = 2π/k. Indeed, the modes with 126 �k = 0 simply describe pure translations of the lattice and have zero stiffness. By conti­ nuity we expect, lim� K̃(�k ) = 0, and ignoring considerations of crystal symmetry, the k→0 expansion of K̃(�k ) at small wavevectors takes the form K̃(�k ) = Bk 2 + O(k 4 ). The odd terms are absent in the expansion, since K̃(�k ) = K̃(−�k ) follows from K(�r −�r ′ ) = K(�r ′ − �r ) in real space. The corresponding frequencies at long wavelengths are ω(�k ) = where v = r Bk 2 = vk, m (VI.40) p B/m is the speed of sound in the crystal. (In a real material, K̃αβ is not proportional to δαβ , and different polarizations of sound have different velocities.) The quanta of vibrational modes are usually referred to as phonons. Using the dis­ persion relation in eq.(VI.40), the contribution of phonons to the internal energy is �Hq � = E0 + X � k,α ¯hvk . −1 eβ¯hvk (VI.41) With periodic boundary conditions in a box of dimensions Lx × Ly × Lz , the allowed wavevectors are �k =  2πnx 2πny 2πnz , , Ly Lz Lx  , (VI.42) where nx , ny , and nz are integers. In the large size limit, these modes are very densely packed, and the number of modes in a volume element d3�k is dN = V dkx dky dkx = d3�k ≡ ρd3�k. 3 (2π) 2π/Lx 2π/Ly 2π/Lz (VI.43) Using the phase space density ρ, any sum over allowed wavevectors can be replaced by an integral as lim V →∞ X f (�k ) = � k Z d3�k ρf (�k ). (VI.44) d3�k h̄vk , 3 β¯ hvk −1 (2π) e (VI.45) Hence, eq.(VI.41) can be re-written as E = E0 + 3V Z B.Z. 127 where the integral is performed over the volume of the Brillouin zone, and the factor of 3 comes from assuming the same sound velocity for the three polarizations. Due to its dependence on the shape of the Brillouin zone, it is not possible to give a simple closed form expression for the energy in eq.(VI.45). However, we can examine its high and low temperature limits. The characteristic temperature separating the two limits, TD = ¯hvkmax ¯ π hv ≈ · , kB kB a (VI.46) corresponds to the high frequency modes at the edge of the zone. For T ≫ TD , all modes behave classically. The integrand of eq.(VI.45) is just kB T , and since the total number R B.Z. 3 of modes is 3N = 3V d �k/(2π)3 , the classical results of E(T ) = E0 + 3N kB T , and C = 3N kB are recovered. For T ≪ TD , the factor exp(β¯hvk) in the denominator of eq.(VI.45) is very large at the Brillouin zone edge. The most important contribution to the integral comes from small k, and the error in extending the integration range to infinity is small. After changing variables to x = βh̄v|�k|, and using d3�k = 4πx2 dx/(βh̄v)3 due to spherical symmetry, eq.(VI.45) gives 3 Z ∞ kB T x3 dx x 4πkB T ¯ e −1 hv 0   3 kT π2 = E0 + V kB T. 10 h̄v 3V lim E(T ) ≈ E0 + 3 T ≪TD 8π  (VI.47) (The value of the definite integral, π 4 /15 ≈ 6.5, can be found in standard tables.) The resulting heat capacity, dE 2π 2 C= = kB V dT 5  kB T ¯ hv 3 , (VI.48) 3 has the form C ∝ N kB (T /TD ) in agreement with observations. The physical inter­ pretation of this result is the following: At temperatures T ≪ TD , only a fraction of the phonon modes can be thermally excited. These are the low frequency phonons with energy quanta h̄ω(�k ) ≪ kB T . The excited phonons have wavevectors |�k| < k ∗ (T ) ≈ (kB T /h̄v). Quite generally, in d space dimensions, the number of these modes is approximately V k ∗ (T )d ∼ V (kB T /¯hv)d . Each excited mode can be treated classically, and contributes roughly kB T to the internal energy which thus scales as E ∼ V (kB T /¯hv)d kB T . The corresponding heat capacity, C ∼ V kB (kB T /¯hv)d , vanishes as T d . 128 VI.C Black-body Radiation Phonons correspond to vibrations of a solid medium. There are also (longitudinal) sound modes in liquid and gas states. However, even “empty” vacuum, can support fluc­ tuations of the electromagnetic (EM) field, photons, which can be thermally excited at finite temperatures. The normal modes of this field are EM waves, characterized by a � = 0 in free space, the elec­ wave-number �k, and two possible polarizations α. (Since ∇ · E tric field must be normal to �k, and only transverse modes exist.) With appropriate choice of coordinates, the Hamiltonian for the EM field can be written as a sum of harmonic oscillators, 1X H= 2 � k,α  p̃�k,α 2 + ωα (�k )2 ũα (�k ) 2  , (VI.49) with ωα (�k ) = ck, where c is the speed of light. With periodic boundary conditions, the allowed wavevectors in a box of size L are �k = 2π(nx , ny , nz )/L where {nx , ny , nz } are integers. However, unlike phonons, there is no Brillouin zone limiting the size of �k, and these integers can be arbitrarily large. The lack of such a restriction leads to the ultraviolet catastrophe in a classical treatment: As there is no limit to the wavevector, assigning kB T per mode leads to an infinite energy stored in the high frequency modes. (The low frequencies are cut off by the finite size of the box.) It was indeed to resolve this difficulty that Planck suggested that the allowed values of EM energy must be quantized according to the Hamiltonian   X 1 q ¯ hck nα (�k ) + H = , with nα (�k ) = 0, 1, 2, · · · . 2 (VI.50) � k,α As for phonons, the internal energy is calculated from q E = �H � = X � k,α ¯hck  1 e−β¯hck + 2 1 − e−β¯hck  2V = V E0 + (2π)3 Z d3�k ¯hck . −1 ¯ eβhck (VI.51) The zero–point energy is actually infinite, but as only energy differences are measured, it is usually ignored. The change of variables to x = βhck ¯ allows us to calculate the excitation energy, h̄c E∗ = 2 V π π2 = 15  kB T h̄c  kB T h̄c 4 Z 0 3 129 ∞ dx x ex − 1 kB T. (VI.52) The EM radiation also exerts a pressure on the walls of the container. From the partition function of the Hamiltonian (VI.50),    Y −β¯hck/2 X 1 e � � Z= exp −βh̄ω(k ) nα (k ) + = , 2 1 − e−β¯hck k )} {nα (� (VI.53) � k,α the free energy is F = −kB T ln Z = kB T = 2V Z 3� d k (2π)3  X  βh̄ck 2 � k,α −β¯ hck + ln 1 − e
 (VI.54) 
¯hck −β¯ hck + kB T ln 1 − e . 2 The pressure due to the photon gas is Z
 ∂F d3�k  −β¯ hck P =− =− ¯ T ln 1 − e hck + 2k B ∂V T (2π)3 Z ∞
kB T = P0 − 2 dk k 2 ln 1 − e−β¯hck (integrate by parts) π 0 Z kB T ∞ k 3 βhce ¯ −β¯hck = P0 + 2 dk (compare with eq.(VI.51)) 3 1 − e−β¯hck π 0 1 E . = P0 + 3 V (VI.55) Note that there is also an infinite zero–point pressure P0 . Differences in this pressure lead to the Casimir force between conducting plates, which is measurable. The extra pressure of 1/3 times the energy density can be compared to that of a gas of relativistic particles. (As shown in the test problems, a dispersion relation, E ∝ |p� |s , leads to a pressure, P = (s/d)(E/V ) in d dimensions.) Continuing with the analogy to a gas of particles, if a hole is opened in the container wall, the escaping energy flux per unit area and per unit time is E . (VI.56) V All photons have speed c, and the average of the component of the velocity perpendicular φ = �c⊥ � to the hole can be calculated as 1 �c⊥ � = c × 4π resulting in φ= Z π/2 2π sin θdθ cos θ = 0 4 4 1 E π 2 kB T c = 3 2 . 4 V 60 h̄ c 130 c , 4 (VI.57) (VI.58) The result, φ = σT 4 , is the Stephan–Boltzmann law for blackbody radiation, and σ= 4 π 2 kB ≈ 5.67 × 10−8 Wm−2◦ K−4 , 60 ¯h3 c2 (VI.59) is Stephan’s constant. Blackbody radiation has a characteristic frequency dependence: Let R E(T )/V = dkE(k, t), where E(k, T ) = ¯hc k3 , π 2 eβ¯hck − 1 (VI.60) is the energy density in wavevector k. The flux of emitted radiation in the interval [k, k+dk] is I(k, T )dk, where 2 3   ckB T k 2 /4π c ¯hc k → I(k, T ) = E(k, T ) = 2 β¯ hck  −1 4 4π e 2 3 −β¯ hck h̄c k e for k ≪ k ∗ (T ) /4π 2 . (VI.61) ∗ for k ≫ k (T ) The characteristic wavevector k ∗ (T ) ≈ kB T /h̄c separates quantum and classical regimes. It provides the upper cutoff that eliminates the ultraviolet catastrophe. 131