H.2 Euler-Maclaurin Sum Formula
H.2.1 Approximate Evaluation of Infinite Sums
I.4 Boson and Fermion Number Operators
Having now established in detail the allowed eigenvalues and eigenvectors ofa†aandaa† for both bosons and fermions, we focus attention on the number operator
Nˆ =a†a, (I.38)
which in both cases has the propertyNˆ|n =n. The only difference is thatn=0, 1, 2, 3,. . . for bosons butn=0, 1 only for fermions. Forbothbosons and fermions,Nˆ satisfies the commutation relations
[ ˆN,a] = −a and [ ˆN,a†] =a†. (I.39) Therefore
ˆ
Na|n =(aNˆ −a)|n =(n−1)a|n (I.40) and
Naˆ †|n =(a†Nˆ +a†)|n =(n+1)a†|n. (I.41) For both bosons and fermions,a|0 ≡ 0, but for fermions, we also havea†|1 ≡ 0. If we regardnas being the number of particles in a state, thenaapplied to|nresults in a state, if such a state|n−1exists, having one less particle. Therefore,ais called an annihilation operator. Similarly, sincea†applied to|nresults in a state|n+1, if such a state exists,
564 THERMAL PHYSICS
having one more particle,a†is called a creation operator. In the case of fermions, and in accordance with the Pauli exclusion principle, a state can have only zero or one particle.
These ideas can be generalized to a number of identical particles whose single-particle states we denote by Greek subscripts. For bosons, the corresponding commutation rela- tions become
[aα,a†β] =δα,β; [aα,aβ] =0; [a†α,a†β] =0. (I.42) The number operator for the single-particle stateαisNˆα = a†αaαand obviously Nˆαand Nˆβ commute and can have a common set of eigenstates. The counterpart to Eq. (I.24) becomes
|nα,nβ,nγ,. . . = 1
(nα!nβ!nγ! ã ã ã)1/2(a†α)nα(a†β)nβ(a†γ)nγã ã ã |0, 0, 0,. . ., (I.43) where the ground state|0, 0, 0,. . .is usually called thevacuum state.
For fermions, the anticommutation relations become
{aα,a†β} =δα,β; {aα,aβ} =0; {a†α,a†β} =0. (I.44) In that case,aαaα = 0 anda†αa†α=0 for allα, but forα = β, we haveaαaβ= −aβaαand a†αa†β = −a†βa†α. For fermions as well,NˆαandNˆβ commute, although the fact that they do is not as obvious as for bosons. The relation corresponding to Eq. (I.43) is simply
|nα,nβ,nγ,. . . =(a†α)nα(a†β)nβ(a†γ)nγã ã ã |0, 0, 0,. . ., (I.45) where the only allowed values of thenαare zero and one. Since these operators anticom- mute, one can order them [8, p. 268] with increasing subscripts (α < β < γ < ã ã ã) to prevent an uncertainty of±1 in the phase.
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Note: Page numbers followed byf indicate figures, bytindicate tables, and bynpindicate footnotes.
A
Absolute activity, 362–363, 371, 405–406, 413–416, 447, 458
Absolute temperature, 4, 7, 13, 20, 32–34, 43, 47, 49, 51f, 53, 121, 124, 156, 247, 259, 262, 266, 289
Absolute zero of temperature, 4–5, 37, 49–51, 425 Acceptors, 443, 446–448, 447f
Accessible quantum state, 49, 313 Activated process, 111–112
Activation energy, 111–112, 181, 392, 395, 440 Activity, 175
absolute, 362, 370–371, 405, 413–416, 447, 458 Actual state, 219
adsorption equation in, 220–221 Adiabatic demagnetization, 329–330f Adiabatic, 25–28, 272
and Carnot cycle, 35, 36f and isentropic process, 272 Bose gas, 423
coefficient of expansion, 506 compressibility, 105, 503 cyclic process, 35 definition, 25np
demagnetization, 329, 330f ideal gas expansion, irreversible,
27–28, 44
ideal gas expansion, reversible, 25–27 reversible contraction, 44
reversible isentropic for ideal second law, entropy change, 34–35 with reversible work, 45
Adsorption equation of Gibbs actual state, 220–221 reference state, 218–219 Affinity, 173–175, 174f
Angular momentum, 537, 539–543 for paramagnetic system, 324 orbital, for electronic structure, 381 polyatomic molecule, 358
probability distribution for, 358 spin, for electronic structure, 381 Anisotropy
of crystal surface or interfacial free energy per unit area, (γ), 196, 215, 221, 223–226, 510–511
illustration for cubic crystal, 223, 242 Annihilation operators, 559–564.
See alsoCreation operators Anti-symmetric fermion states, 465–466 Anti-symmetrization operator, 467 Approximate
evaluation of infinite sums, 556
evaluation of partition function (thermody- namic perturbation theory), 549–552 Arrhenius form, 112f–113, 181
Asymptotic, 277–278, 304f, 422, 449 expansion, 409–410, 435
series, 499–501, 553–556
series,vs.convergent series, 500–501
Avogadro’s number, 3, 12, 48, 49, 126–127, 166, 289
B
Band gap, 425, 442, 443f, 445
Beale, Paul, xv, xviii, 6, 350, 416, 419, 483, 489 Bernoulli numbers and polynomials, 553–554 Binary liquid, in gravity, 162–164
Binary solutions
chemical components, 137 chemical potentials, 137–138 chord construction, 141
569
570 Index
Binary solutions(Continued) Euler equation, 137–138 general solution, 153
Gibbs-Duhem equation, 137–138 graphical constructions, 139–141 ideal, 142–145
ideal solid and ideal liquid, 145–148 ideal solutions, 142–145
intercept and common tangent constructions, 139–141
lens type binary phase diagram, 145–146, 147–148
liquid in gravitational field, 162–164 miscibility gap, 137–141, 142–148, 153 molar Gibbs free energy, 139
mutual solubility, 144
phase diagram, ideal solid-liquid, 145–148
regular solution, 148–152 thermodynamics of, 137–141
Binary system,see alsoMulticomponent system partial molar quantities, 3, 74f, 75 Blackbody radiation, 298–302
Bohr magneton, 324–325, 437–439 Boltzmann, Ludwig, 251–252
constantkB, 12, 48, 49, 249–251 distribution, 285–289
equation, 252–253 Eta theorem, 247, 251–256
factor, 250, 263, 313, 322np, 337, 361, 373, 468, 484–485, 487
sampling, 487 Boltzons, 468
Bomb calorimeter, 168 Bose, Satyendra
condensation, 413
chemical potential belowTc, 413 condensate fraction, 416f condensate region, 421–424 critical temperatureTc, 421 entropy belowTc, 417–418 heat capacity, 419f, 420–421 internal energy belowTc, 417–418 λpoint, heat capacity, 420–421 pressure belowTc, 417
region inv, pplane, isotherms, 422–423 region inv, Tplane, isotherms, 421–422 thermodynamic functions, 416–421 -Einstein distribution function, 374–375 ideal gas, 376–378, 410–412
Bosons, 465–468
at low temperatures, 413–416 operators, 560–562
number operators, 563–564
Bragg-Williams molecular field approximation, 471np
Brillouin function, 325–326 Bromwich contour, 330–331 Bulk samples, homogeneous, 6–7 C
Cahn, John W., 185
determinants for surface invariants, 194–197
layer model, 185, 192–197 Calorie, 17
Canonical distribution, 337np
Canonical ensemble, 288, 457–458.See also Classical canonical ensemble Boltzmann factor, 337
canonical ensemble, classical
averaging and equipartition, 343–345 canonical transformations, 354–356,
529–535
effusion of ideal gas, 340–342
Maxwell-Boltzmann distribution, 338–339 classical ideal gas, 313–316, 338–342
Maxwell-Boltzmann distribution, 338–339 quantum concentration, 339–340
definition, 305
density of states, 330–331
derivation from microcanonical, 305–312, 360–368
energy dispersion, fluctuation, 320–321, 367–368
factorization theorem, 312–313
grand,seeGrand canonical ensemble (GCE) Helmholtz free energy, 306np, 307–309, 311,
315–316, 321–322
Maxwell-Boltzmann distribution, 317–319
as most probable distribution, 309–312 paramagnetism, 290–292, 321–330
adiabatic demagnetization, 329–330 classical treatment, 322–324 magnetic moment, 321–325 properties, 327–329
quantum treatment, 324–327
particles, negligible interaction energies, 285, 312–313
blackbody radiation, 298–302 harmonic oscillator, 293–302 heat capacity of a crystal, 297–298 rigid linear rotator, 303–304 two-state subsystems, 289–293 partition function, 330–331
diatomic molecular gas, 382–387 polyatomic molecule, 387–388 relation to density of states, 330–331 relation to Helmholtz free energy, 311,
315–316, 321–322
virial expansion coefficients, 348–354 virial theorem for time averages, 346–348 Canonical partition function, 457–458
for single spin, 471–472 Canonical transformations
general transformation, 529–530
Jacobian value, 354–356, 529–530, 532–533 necessary and sufficient conditions, 530–534 restricted transformation, 534–535
symplectic transformation, 532–534 use of, 354–356
Canonical variables for freely rotating poly- atomic molecule, 546
Capillary
length, 201, 205–206, 211 rise in tube, 185, 200, 200f Carnot, Sadi, 35–38
cycle, 4, 32np, 35–36 efficiency, 36–37 engines, 35–38, 36f refrigerator, 38
Cauchy stress tensor, 216–217, 218–219 Celsius scale, 4np
Chemical heat, 53
Chemically closed system, 15, 32, 33, 41, 53, 75, 84, 89–90, 167, 168–169, 173, 174
Chemical potential, 12, 53–54, 55, 61, 64–66, 69, 83, 91–93
of binary solutions, 137, 139np, 140f, 142–143, 145, 153
Bose gas below critical temperature, 413 electrochemical, 12, 155, 166
gravitational, 12, 155, 157–158 ideal gas, 54, 61
intrinsic, 12, 157–158 monatomic gas, 55
of monocomponent ideal gas, 54, 161
multicomponent systems, 53, 55, 171–172, 175–176
real gases, 64, 171, 176 Chemical reaction
affinity, 173, 174f, 182 among ideal gases, 177, 178 at constant volume/pressure, 168 δG(G) of, 173
δH(H) of, 170
entropy production during, 75, 77, 174–175 equilibrium, 173–175
equilibrium conditions, explicit
equilibrium constant,K, 176, 176np, 178–181 alternative,Kc
dependence on pressure, 182 dependence on temperature, 180 extension of equilibrium conditions to
include, 93 heat of, 170
heterogeneous solids/liquids with gases, 171, 179
in isolated system, 167
reaction product and quotient, 176–177 simultaneous reactions, 182–183 standard states, 171–173
stoichiometric coefficients, 76, 167 Chord construction, 129–130, 129f, 141,
142f, 153 binary solution, 141
Classical canonical ensemble, 337.
See alsoCanonical ensemble
averaging theorem and equipartition, 343
572 Index
Classical canonical ensemble(Continued) classical ideal gas, 345
Cartesian coordinates, 340–341 effusion, 341, 342
Maxwell-Boltzmann distribution, 338–339 law of Dulong and Petit, 342–343, 345 rotating rigid polyatomic molecules, 356–358 use of canonical transformations, 354–356 virial coefficients, 348, 353
virial theorem, 346–348 Classical ideal gas, 338–342
canonical ensemble, 313–316 free particle in box, 314–316
grand canonical ensemble, 375–378, 380–388 Classical microcanonical ensemble.See also
Microcanonical ensemble classical harmonic oscillators in three
dimensions, 282, 283–284 classical ideal gas, 281–283 definition, 277
description, 280
Classical particles, Monte Carlo simulation, 491–494
Classical partition function, 337–338 evaluation, 342–343, 354
for single diatomic molecule, 355 Classical treatment of paramagnetism,
322–324 Clausius, Rudolf, 17
-Clapeyron equation, 110–115 first recognition of entropy postulate of, 31–34, 37 Closed system, 15
Coefficients of curvatures, 201–202 Coexistence curves, 109
Common tangent construction, 127–129, 129f of binary solutions, 139–141
Communication theory, 247 Composition, mole fractions, 62 Concave function, 95–96, 100 Concentrations, 64
Condensate region, Bose condensation inv, pplane, 422–423, 423f
inv, Tplane, 421–422, 422f
Conditions for equilibrium of composite systems, 81–83, 87
multicomponent subsystems, 81–83 mutual equilibrium, 93
extension to chemical reactions, 93 Conduction band, 442–444, 446, 447f Configuration
distinguishable particles, 285–286 MC simulation, 484–491
Conjugate variable, 47, 67–68 Conservative external forces, 155 Constant pressure
chemical reactions at, 168–170 heat capacity at, 20, 22–23, 420–421 Constant
Boltzmann, 12, 48, 49, 249–251 Curie, 322–324
equilibrium, 175–182 ideal gas, 4
Planck, 55, 60, 68–69 Stefan-Boltzmann, 301 van der Waals fluid, 126–127 Constant volume
chemical reactions at, 168–170 heat capacity at, 19–20, 22–23 Constrained equilibrium, 217 Construction, graphic
chord binary, 141
monocomponent, 129–130 common tangent
binary, 139–141
monocomponent, 127–129 intercept, 139–141
Maxwell equal area, 133–134, 133f Contact plane, 234–235
Convex function, 100–102 Correlations function
for hard-sphere gas, 492 Monte Carlo, 485 of spin, 484, 485
Creation and annihilation operators.See also Annihilation operators
boson and fermion number operators, 563–564
boson operators, 560–562 eigenstates of, 559, 561–562 fermion operators, 562–563 for harmonic oscillator, 559–560 vacuum state, 564
Critical exponent, 474–475 Critical point, 109
Critical temperature, 421–422 definition, 413, 471–472 Crystal
heat capacity of, 297–298
equilibrium shape, 215–216, 227–228, 233 Crystalline solids, 215–216
Curie constant, 322–324 Curie’s law, 292f, 322–324 Curie-Weiss law, 475–476 Curved interfaces in fluids
capillary length, 200 constants, 198
contact angle, 204–205
dividing surface in comparison system, 197 Gibbs coefficients of curvatures, 201–202 interface junctions and contact angles,
202–205
surface of tension, 198
Curved solid-fluid interfaces. See also Planar solid-fluid interfaces
description, 227–228
discontinuous derivatives ofγ, 228–232 invertedγ-plot, 232–233
D
Decimation, 488–489 de Donder, Théophile, 173np
concept of affinity, 173 Degeneracy factor, 402 Degrees of freedom, 7–8, 55–56 Density
dispersion, fluctuation, 367 matrix, 451–452
one-dimensional harmonic oscillator, 460–461
single free particle, 459–460 spin 1/2 particle, 461–465 operator, 451
assumption of random phases, 454 free particle, 459–460
grand canonical ensemble, 458 harmonic oscillator, 460–461 pure quantum state, 451–452 relationship to entropy, 47–48 statistical quantum state, 453 in terms of Pauli spin matrices,
461–462
time evolution, 455–456 various ensembles, 456–458 of states
canonical ensemble, 330–331 definition, 330–331, 332
Detailed balance, principle of, 486–487 Diatomic molecular gas, 382
heteronuclear molecules, 382–385 homonuclear molecules, 385–387 polyatomic molecular gas, 387–388 Diatomic molecule
moments of inertia, 303 quantum energy levels, 547 rigid linear rotator, 303 Dirac, Paul, 430
continuous spectrum, 452np delta function, 30, 430
Fermi-Dirac distribution, 373–374, 428–432 in semiconductors, 444np
thermionic emission, 439 vector space, 468
Boson and Fermion number operators, 563–564
Boson operators, 560–562 eigenbras and eigenkets, 560–562 Fermion operators, 562–563 probability density ket, 51
Discontinuous derivatives ofγ, 228–232 Disorder function, entropy, 247–251 Distinguishable particles, with negligible
interaction
blackbody (hohlraum) radiation, 298–302 canonical ensemble, 288
configuration, 285–286
crystal, heat capacity of, 297–298
derivation of Boltzmann distribution, 285–289
574 Index
Distinguishable particles, with negligible interaction(Continued)
ensemble, 288
factorization theorem, 312–313 harmonic oscillators, 293–302 identical but, 285
magnetic moment, 290–292 paramagnetism, 290–292 partition function, 286–287 rigid linear rotator, 303–304 Stirling’s approximation, 287 two-state subsystems, 289–293 Divacancies, 393–394
Donors, 442–443, 446–447 Dopants, 442–443, 446–449 E
Effusion
definition, 340–341 energy flux with, 341
of ideal classical gas, 340–342 Eigenstates, 559, 561–562 Einstein, Albert
Bose-Einstein distribution, 374–375 heat capacity, 297
nuclear reactions, 167
quotation re thermodynamics, xvii temperature, 297
Electric fields, external forces, 166
Electronic heat capacity, 381–382, 432–433.
See alsoHeat capacity
Elementary kinetic theory, of gases, 12–13 Elementary method, 498, 504–507 Endothermic reaction, 170 Energy, 3
criterion, 84–88
and entropy, equivalence, 87–88 local energy criterion, 86–87 dispersion
canonical ensemble, 320–321 grand canonical ensemble, 367–368 free, Gibbs, 69
free, Helmholtz, 68–69 internal, 11, 15–16
kinetic of center of mass, 11
lack of partitioning, 19 mechanical, 8–12 single particle
in one dimension, 8–9 in three dimensions, 9–10 as state function, 15–16 system of particles, 10–12 Ensembles, 257, 288
applied to point defects, 391–393 averages, 5–6
canonical, 457–458 grand canonical, 458 microcanonical, 457 pressure, 360, 389–390 Enthalpy, 28–29, 69
of chemical reaction, 75–78 criterion, 90–91
of melting, 29, 30
of multicomponent system, 62–63 of phase change, 46f
stability requirements for, 102–103 Entropy, 5, 31
of Bose condensation, 418, 419 change calculation, 39
change due to heat transfer, 32 change for adiabatic process, 25np change for reversible path, 38 of chemical reaction, 75–78 criterion for equilibrium, 32, 79–84
equivalence to energy criterion, 87–88 Gibbs phase rule, 83, 93, 109, 141 disorder function, 247–251
disorder measurement, 247–251 elementary relationship to
microstates, 47–48 formula, 400–402
for general ensemble, 397–398 example of maximization, 399–400 summation over energy levels, 402–403 ideal gas, 44
information theory, 247–256 of mixing, ideal gases, 275–276 of phase change, 46
probability of microstate, 47–48 relationship to microstates, 47–48
stability requirements for, 95–100 as state function, 31, 32
statistical interpretation, 47–48 of two systems, 250
Equations of state, 20, 23–24, 28, 41–43, 54, 61, 70, 98, 121, 138–139, 250
Equilibrium, 3
chemical reaction, 173–175 condensed phases, 175–176, 175np conditions for subsystems, 81–83 constant for chemical reaction, 176–178,
180
criteria, 79–81, 84–93
dependence on pressure, 182–183 dominant contributions, 525–526 enthalpy criterion, 90–91
entropy additivity, 523, 526–527 entropy criterion, 32, 79–84 explicit conditions for, 175–182 with external forces, 155–157 Gibbs free energy criterion, 89–90 in gravitational field, 157–164
Helmholtz free energy criterion, 88–89 heterogeneous reactions in gases, 179 internal energy criterion, 88
Kramers (grand) potential criterion, 91–92
multiplicity function, 523–524
of two-state systems, detailed study, 523-527
overlap integral, 523 phase rule, 83–84
pressure, dependence ofK(T, p), 182 reactions in gases, 177, 178
rotating systems, 164–166 shape, 227–228
of crystal, 215–216, 227–228 globalvs.local, 239
Legendre transforms, 241–242 fromξ-vector, 236–239 state, macroscopic systems, 3 Summary, 92t
temperature, dependence ofK(T, p), 180, 181
Equimolar surface, 188 Equipartition, 263
averaging theorem and, 343–345 principle, 343
Ergodic hypothesis, 260 Eta theorem, 251–252, 254–256 Euclidean geometry, 6
Euler equation, 60–62, 68, 72, 110–111, 137–138, 158, 169, 193, 201–202, 216–217, 322, 365, 390, 400, 419, 543
Euler-Maclaurin sum formula, 437–439, 554–556
Euler theorem, 59–60
for extensive functions, 60–62 of homogeneous functions, 59–64 for intensive functions, 63–64 Excited states
concentration of particles, 414–415 function ofT/Tc, 416f, 419
Exothermic reaction, 170
Extensive functions, Euler theorem for, 60–62 Extensive variables, 7
External forces
binary liquid, 162–164 centrifuge, 165–166
conditions for equilibrium, 155–157 electric fields, 166
electrochemical potentials, 155, 166 gravitational segregation, 161–162 inhomogeneous pressure, 155 Lagrange multipliers, 156
multicomponent ideal gas, 160–162 non-uniform gravitational field, 164 rotating systems, 164–166
uniform gravitational field, 157–164 Extrinsic semiconductors, 442–443 F
Faceting, of large planar face, 233–235, 233f
Factorization
for independent sites, 370–373 theorem, 312–313
Fahrenheit scale, 4 Fan of vectors, 228–229
576 Index
Fermi, Enrico
degenerate gas, 425 energyF, 428–429 heat capacity, 432–433
Sommerfeld expansion, 430–432 sphere inkspace, 433–434 temperatureTF, 427
thermal activation, electrons, 429–433
thermionic emission of electrons, 439–442
wavenumberkF, 426
-Dirac distribution function, 373–374, 428, 429, 430f, 432
energy, 427, 428–429, 431–432, 433–434, 439, 443–444
ideal gas, 376–378, 410–412 level, 432
sphere, 426–427 wavenumber, 426–427 Fermion operators, 562–563
number operators, 563–564
Fermions, 425–450, 467–468.SeeBosons First law of thermodynamics
combined with second law, 41–47 discussion of, 16–17
enthalpy, 28–29 heat capacities, 19–23 ideal gas expansion, 24–28 quasistatic work, 17–19 statement of, 15–17 Fluid-fluid interfaces
contact lines, 202np, 207 curved interfaces, 197–202
interface junctions and contact angles, 202–205
planar interfaces in, 186–197 sessile drops, 185–186, 210–211 surface shape in gravity, 205–213 three-dimensional problems, 210–213 two-dimensional problems, 206–209 Forces, external
conservative, equilibrium condition, 155 electrical, 166
non-uniform gravitational, 164
for rotating systems, 164–166 uniform gravitational, 157–164 Frenkel defects, 395
Fugacity, 64–67, 65f
ratio, chemical reactions, 175–176 Functionshv(λ, a), 408–410, 410f Fundamental equation of system, 42, 43 Fundamental hypothesis, 258
statistical mechanics, 258–260 G
Gamma function, 499, 500f Gamma-plot (γ-plot), 227, 228f
discontinuous derivatives of, 228–232 inverted gamma-plot, 232–233 minimum gamma plot (-plot),
234–235, 235f
Gauss divergence theorem on a surface, 515–516
Gaussian
approximation, 524–525 curvature, 512–513 distribution, 317–319, 461 integral, 459–460
GCE.SeeGrand canonical ensemble (GCE) General ensemble, entropy for, 397–398
example of maximization, 399–400 summation over energy levels, 402–403 Giauque, William, xvi
Gibbs, J. Willard
adsorption equation, 190–192, 215–216 boltzon weighting factor, 468
coefficients of curvatures, 201–202 correction factor for extensivity, 268–271 correction factor, monatomic ideal gas with,
268–271 distribution, 305
dividing surface, 185, 186f, 187–190 -Duhem equation, 61–62, 109–110, 218–219 factor, 360–361
free energy, 69, 173, 174f binary solutions, 139 equilibrium criteria, 89–90 stability requirements for, 103–104 van der Waals fluid, 129–130