Basic Theoretical Physics: A Concise Overview P42 potx

428 55 Conclusion to Part IV For a grand canonical ensemble, where the heat bath not only exchanges energy with the system, but also particles, such that the particle number in avolumeelementV fluctuates around the average N(T,μ,V), in addition to β  = 1 k B T  , one obtains for the distribution a second parameter μ (the so-called chemical potential ) for the analogous quantity to the Helmholtz free energy, i.e., for the Gibbs grand canonical thermodynamic potential: Φ(T,μ,V, )=−k B T · ln Z(T,μ,V, ) , with the grand canonical partition function: Z(T,μ,V, )=  i,j e − E i (V,N j )−μN j k B T . The mathematical relation between the Helmholtz free energy and the Gibbs grand canonical potential Φ is a Legendre transform, i.e.: Φ(T,μ,V, )=F (T,V,N(T,μ,V), ) − μ · N(T,μ,V) , similarly to the way the internal energy U(T,V,N, )andtheenthalpy I depend on each other: I(T,p,N, )=U(T,V(T,p,N, ),N, )+p · V (T,p,N, ) , with the pressure p as the conjugate Lagrange parameter regulating fluctua- tions in V . These Legendre transformations are mathematically analogous to the transition from the Lagrange function L(v, ) 2 to the Hamilton function H(p, ) in classical mechanics; incidentally (this may be used for mnemonic purposes!) the corresponding letters are similar, i.e., V (and v)andp (and p), although the meaning is completely different. The relation between a) and b) can also be expressed as U(T,V,N, ) ≡H(V,N, ) T . Where  ˆ A  T is the thermodynamic expectation with the suitable canoni- cal (and microcanonical and grand canonical ) Boltzmann-Gibbs distribution, e.g.,  ˆ A  T =  i p i (T ) ·ψ i | ˆ A|ψ i  , with p i (T )= e − E i (V,N, ) k B T Z(T,V,N, ) , 2 Actually by the Legendre transformation of −L. 55 Conclusion to Part IV 429 and the Hermitian operator ˆ A represents an observable (i.e., a measurable quantity). The ψ i represent the complete system of eigenfunctions of the Hamilton operator H;theE i are the corresponding eigenvalues. Concerning the entropy: This is a particularly complex quantity, whose complexity should not simply be “glossed over” by simplifications. In fact, the entropy is a quantitative measure for complexity, as has been stated already. In this context one should keep in mind that there are at least three commonly used methods of calculating this quantity: a) by differentiating the Helmholtz free energy with respect to T : S(T,V,N, )=− ∂F(T,V,N, ) ∂T ; b) from the difference expression S = 1 T ·(U − F ) following from the relation F (T,V,N, )=U(T,V,N, ) −T · S(T,V,N, ) , where the quantity T ·S represents the heat loss. This formulation seems to be particularly useful educationally. c) A third possibility of quantification follows directly from statistical physics S = −k B ·  i p i ln p i (where this relation can even be simplified to S = −k B ·ln ˆ ˆ in the above trace formalism.) d) Shannon’s informational entropy 3 is also helpful. It should also have become clear that the Second Law (and even the Third Law) can be formulated without recourse to entropy. However, the notion of absolute temperature (Kelvin temperature) T is indispensable; it can be quantified via the efficiency of Carnot machines and constitutes a prerequisite for statistical physics. Amongst other important issues, the Maxwell relations remain paramount. Here one should firstly keep in mind how these relations follow from a differ- ential formulation of the First and Second Laws in terms of entropy; secondly one should keep in mind the special relation ∂U ∂V = T ∂p ∂T − p 3 This is essentially the same: k B is replaced by 1, and the natural logarithm is replaced by the binary logarithm. 430 55 Conclusion to Part IV and the application to the Gay-Lussac experiment, and thirdly one should remember that one can always obtain important cross-relations by equating mixed second-order derivatives, e.g., ∂ 2 F ∂x i ∂x j , in the total differentials of the Helmholtz free energy, dF = −pdV + μdN + −SdT, or analogous thermodynamic potentials. In this book we have also stressed similarities between the four different parts. Therefore, looking back with a view on common trends, it seems that the systematic exploitation of “coherence properties” has a promising future, not only in optics (holography), but also in quantum mechanics (quantum computing, etc.). Unfortunately, as discussed above, thermalization also leads to decoherence. However, recent success in obtaining ultralow temperatures means that this barrier may become surmountable in the not-too-distant future. References 1. W. Nolting: Grundkurs theoretische Physik, in German, 5th edn (Springer, Berlin Heidelberg New York 2002), 7 volumes. This is a good example of a series of textbooks which in seven volumes covers sometimes less but sometimes much more than the four parts of our com- pendium. 2. http://www.physik.uni-regensburg.de/forschung/krey To Part I: 3. As a recommendable standard textbook on Classical Mechanics we recommend H. Goldstein, Ch.P. Poole, J. Safko: Classical Mechanics, 3rd edn (Addison- Wesley, San Francisco Munich 2002), pp 1–638 4. H. Hertz: Die Constitution der Materie,in:A.F¨olsing (Ed.), Springer, Berlin Heidelberg New York, 1999, pp 1–171 5. A. Einstein: Zur Elektrodynamik bewegter K¨orper, Ann. der Physik 17, 891 (1905) 6. R. von E¨otv¨os, D. Pek´ar, E. Fekete: Ann. d. Physik 68, 11 (1922) 7. S.M. Carroll: Spacetime and Geometry. Introduction to General Relativity,(Ad- dison Wesley, San Francisco and elsewhere 2003), pp 1–513 8. L.D. Landau, E.M. Lifshitz: Course of Theoretical Physics, volume 2, The Clas- sical Theory of Fields, 4th edn (Pergamon Press, Oxford New York and else- where, 1975), pp 1–402 (This book contains very readable chapters on relativ- ity.) 9. H.G. Schuster, W. Just: Deterministic Chaos, 4th edn (Wiley-VCH, Weinheim 2005), pp 1–287 To Part I I: 10. B.I. Bleaney, B. Bleaney: Electricity and Magnetism, 3rd edn (Oxford Univer- sity Press 1976), pp 1–761 11. J.D. Jackson: Classical Electrodynamics, 3rd edn (Wiley, New York Weinheim Singapore 1999), pp 1–938 12. A. Heck: Introduction to Maple, 3rd edn (Springer, Berlin Heidelberg New York 2003), pp 1–828 13. L.D. Landau, E.M. Lifshitz: Course of Theoretical Physics, volume 1, Mechan- ics, 3rd edn (Pergamon Press, Oxford New York and elsewhere, 1976), pp 1–169, Chapt. 44 . (This chapter is useful for understanding the cross relations to the Fermat principle in geometrical optics.) 14. A. Sommerfeld: Optics, 4th edn (Academic Press, New York 1967), pp 1–383 432 References 15. F. Pedrotti, L. Pedrotti: Introduction to optics, 2nd edn (Prentice Hall, Upper Saddle River (NY, USA) 1993), pp 1–672 16. E. Hecht: Optics, 4th edn (Addison-Wesley, San Francisco New York 2002), pp 1–565 17. K. Bammel: Physik Journal, in German, (1) 42 (2005) To Part III: 18. A. Einstein: ¨ Uber einen die Erzeugung und Verwandlung des Lichtes betref- fenden heuristischen Gesichtspunkt, Ann. der Physik 17, 132 (1905) 19. W.Heisenberg:Z.Physik33, 879 (1925) 20. M. Born, W. Heisenberg, P. Jordan: Z. Pysik 35 557 (1926) 21. C.I. Davisson, L.H. Germer: Nature 119, 890 (1927); Phys. Rev. 30, 705 (1927) 22. E. Schr¨odinger: Ann. Physik (4) 79, 361; 489; 734 (1926); 80, 109 (1926) 23. J. von Neumann: Mathematische Grundlagen der Quantenmechanik,inGer- man, reprinted from the 2nd edn of 1932 (Springer, Berlin Heidelberg New York 1996), pp 1–262 24. W. D¨oring: Quantenmechanik, in German (Vandenhoek & Ruprecht, G¨ottingen 1962), pp 1–517 25. J. Bardeen, L.N. Cooper, J.R Schrieffer: Phys. Rev. 106, 162 (1957); 108, 1175 (1957) 26. D.D. Osheroff. R.C. Richardson, D.M. Lee, Phys. Rev. Lett. 28, 885 (1972) 27. H. B¨orsch, H. Hamisch, K. Grohmann, D. Wohlleben: Z. Physik 165, 79 (1961) 28. M. Berry: Phys. Today 43 34 (12) (1990) 29. A.Einstein,B.Podolski,N.Rosen:Phys.Rev.47, 777 (1935) 30. J.S. Bell: Physics 1, 195 (1964) 31. P. Kwiat, H. Weinfurter, A. Zeilinger: Spektrum der Wissenschaft 42, (1) (1997) 32. A. Zeilinger: Einsteins Schleier – die neue Welt der Quantenphysik, in German (C.H. Beck, M¨unchen 2003), pp 1–237 33. D. Loss, D.P. DiVincenzo: Phys. Rev. A 57, 120 (1998) 34. F.H.L. Koppens, J.A. Folk, J.M. Elzerman, R. Hanson, L.H.W. van Beveren, I.T. Fink, H.P. Tranitz, W. Wegscheider, L.M.K. Vandersypen, L.P. Kouwen- hoven: Science 309, 1346 (2005) To Part IV: 35. A. Einstein: ¨ Uber die von der molekularkinetischen Theorie der W¨arme geforderte Bewegung von in ruhenden Fl¨ussigkeiten suspendierten Teilchen, Ann. der Physik 17, 549 (1905) 36. P. Papon, J. Leblond, P.H.E. Meijer: The Physics of Phase Transitions (Springer, Berlin Heidelberg New York 2002), pp 1–397 37. C. Kittel: Introduction to Solid State Physics, 8th edn (Wiley, New York London Sidney Toronto 2005), pp 1–680 38. W. Gebhardt, U. Krey: Phasen¨uberg¨ange und kritische Ph¨anomene, in German (Vieweg, Braunschweig Wiesbaden 1980), pp 1–246 39. W. D¨oring: Einf¨uhrung in die theoretische Physik, Sammlung G¨oschen,inGer- man, 5 volumes, 3rd edn (de Gruyter, Berlin 1965), pp 1–125, 1–138, 1–117, 1–107, 1–114 40. A. Sommerfeld, H.A. Bethe: Elektronentheorie der Metalle (Springer, Berlin Heidelberg New York 1967) References 433 41. H. Thomas: Phase transitions and critical phenomena. In: Theory of condensed matter, directors F. Bassani, G. Cagliotto, J. Ziman (International Atomic Energy Agency, Vienna 1968), pp 357–393. At some libraries this book, which has no editors, is found under the name E. Antoncik. 42. R. Sexl, H. Sexl: White dwarfs – black holes, 2nd edn (Springer, Berlin Heidel- berg New York 1999), pp 1–540 43. R. Sexl, H.K. Urbantke: Gravitation und Kosmologie,inGerman,3rdedn(Bib- liograpisches Institut, Mannheim 1987), pp 1–399 44. C.W. Misner, K.S. Thorne, J.A. Wheeler: Gravitation, 25th edn (Freeman, New York 1003), pp 1–1279 45. D. Vollhardt, P. W¨olfle, The superfluid phases of helium 3,(Taylor&Francis, London New York Philadephia 1990), pp 1–690 46. V.L. Ginzburg, L.D. Landau: J. Exp. Theor. Physics (U.S.S.R.) 20, 1064 (1950) 47. A. Abrikosov: Sov. Phys. JETP 5, 1174 (1957) 48. L.D. Landau, E.M. Lifshitz: Course of Theoretical Physics, volumes 5 and 9 (= Statistical Physics, Part 1 and Part 2), 3rd edn, revised and enlarged by E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford New York and elsewhere, 1980), pp 1–544 and 1–387 49. J. de Cloizeaux: Linear response, generalized susceptibility and dispersion the- ory. In: Theory of condensed matter, directors F. Bassani, G. Cagliotto, J. Ziman (International Atomic Energy Agency, Vienna 1968), pp 325–354. At some libraries this book, which has no editors, is found under the name E. Antoncik. 50. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller: J. Chem. Phys. 21, 1087 (1953) 51. D.P. Landau, K. Binder: A guide to Monte Carlo simulations in statistical physics, 2nd edn (Cambridge University Press, Cambridge UK, 2000), pp 1– 448 Index Abb´e resolution 198 Abrikosov (vortex lattice) 398 Abrikosov, Ginzburg, Legget 395 abstract quantum mechanics (algebraic methods) 241 accelerated reference frames 95 acceleration (definition) 11 acceleration in planar polar coordinates 31 acceptance (Metropolis algorithm) 412 accuracy for optical mappings 197 action and reaction : weak and strong forms 8, 9 functional 47 principle 55 and reaction . . . 8, 38, 119 activated state (thermodynamics, k B T  E i ) 341 active charge 119 active gravitational mass 8 actual (versus virtual) orbits 47, 48 addition rules for angular momenta 255, 270 additivity of partial entropies 415 additivity of the entropy 365, 413, 414 adiabatic expansion 324 demagnetization 372 demagnetization (low temperatures) 371 adiabatics versus isotherms 325 admixture of excited states (“polariza- tion”) 262 aether (pre-Einstein) 58 Aharonov-Bohm effect 281, 287, 295 algebraic methods (in quantum mechanics) 241 Alice (quantum cryptography) 289, 297 alkali atoms 256 alternating parity 226 Amp`ere’s law 145 current loops (always equivalent to magnetic dipoles) 149 law 145, 156 law including Maxwell’s displacement current 153 amplitude resonance curve 20 angular momentum operators (orbital part) 235 elementary treatment 24 of a rigid body 73, 77 orbital part, spin part 236 quantization 351 quantum number l 237 anholonomous constraints 45, 89 antiferromagnetism 275 aperture (rectangular, circular, Fraunhofer diffraction) 198 aphelion (as opposed to perihelion) 34 arc length 12 archive kilogram 7 area velocity 38 Arrott’s equation (magnetism) 330 artificial atoms (quantum dots) 285 ascending ladder operator 243 aspects of relativity (Part I, Part II) 301 asymmetric heavy top 84 autonomous 86 Avogadro’s law (of constant propor- tions) 303, 335 436 Index axial (versus polar) vectors: (v 1 × v 2 ) 71 azimuth ϕ 126 ballistically driven oscillation 18 Bardeen, Cooper, Schrieffer (BCS) 277, 395 barometric pressure 352 barrier (tunneling) 229 basic quantities 7 battery 127 battery voltage 128 BCS theory (Bardeen, Cooper, and Schrieffer) 277, 395 beats 183 Bell 279, 281, 295 Bell experiments 281 Bernoulli’s general pressure formula 335, 336, 338, 384 Berry phases 281, 282 biaxial crystals (anomalous birefringence) 190 versus uniaxial crystals (birefrin- gence) 191 Binet ellipsoid 78 biological danger (of the energy impact of high-frequency radiation fields) 171 Biot and Savart 148, 149 birefringence crystal optics 188 Fresnel ellipsoid versus index ellipsoid (E versus D) 192 phase velocity versus ray velocity 192 black holes (stars) 97, 387 black-body radiation laws 207 blackening function (of a photographic plate) 200 Bob (quantum cryptography) 289, 297 Bohr ’s atomic model 209 ’s magneton μ B 250, 406 -Sommerfeld quantization 209 ’s atomic radius 237 (Copenhagen interpretation of quantum mechanics) 220, 295 general remarks (if at all) 4, 209 bolometry 207, 347 Boltzmann 414 ’s constant k B 184, 302 -Gibbs distribution (canonical ensemble) 337, 343, 344, 427 -Gibbs distribution (grand canonical) 367, 381, 394, 428 probabilities 302 statistics 381 Born 209, 220 Bose and Fermi gases 335, 337, 379 Bose and Fermi statistics 337 Bose-Einstein condensation 278, 371, 374, 375, 391 bosons 252 bosons and fermions 379, 383 bound states 225 bound systems in a box 224 boundary currents (for superconductors) 146 divergence 138 rotation 138 value problem (heat conduction) 307 Boyle-Mariotte law (ideal gas) 324, 335 brane theories (formal membranes etc.) 4 bucket experiment 98 building up of the oscillation amplitude 69 C V = γk B T (in metals) 385 caloric (versus thermal) equation of state 317 calorie (heat) 306 canonical and grand canonical ensembles 366 angular-momentum commutation relations 243 ensemble (Boltzmann-Gibbs) 337 momentum (as opposed to the kinetic momentum) 61 and grand canonical ensembles 366 commutation relation 216 ensembles (microcanonical, canoni- cal, grand canonical) 367 equations of motion 53, 248 Index 437 transformations 53 ensemble (Boltzmann-Gibbs) 427 capacities (plate, sphere, cylinder) 131 capacitor 127 capacity 127, 131 Cardani suspension 81, 82 Carnot coordinates 358 heat engines 355 process 355, 358 process, infinitesimal (Clausius- Clapeyron) 370 cat states (Schr¨odinger) 221 Cauchy problem (initial values, heat diffusion) 307, 311 causality 21 cause and effect 169 center of mass, definition 23 center of mass, theorem 29 centrifugal force 39, 96 centripetal acceleration 12 cgs system 7 chaos 4, 80, 85, 86 charge density (general) 114 active and passive 119 density (true versus effective) 165 charged superfluid 395 chemical phase equilibria 413 potential μ 314, 330, 338, 346, 365, 366, 379, 392, 393, 418, 428 potential μ R for a droplet 425 chirality 287 chromodynamics 120 circulation 116 classical ideal gas 337, 339 classical mechanics (Part I) 301 classification of Minkowski four-vectors: space-like, light-like, time-like 101 Clausius -Clapeyron equations 305, 369 impossibility of an ideal heat pump 355 pressure in interacting systems 341 Clebsch-Gordan coefficients 256 closed systems 363 co-moving clock 96 with the particle (relativistic) 55 cartesian vectors for a rigid body 79 clock 59 coexistence region Arrott 331 van der Waals 328, 329 coherence length ξ(T ) (superconductiv- ity) 397 coherence length and time 199 coherent superposition 199, 283, 285, 291, 295 coherent superposition (Schr¨odinger’s cat) 220 comets 37 commutation relation (canonical) 216 commutator (quantum mechanics) 53, 294 complexity 315 complexity (quantitative measure: entropy S) 429 composition of angular momenta 255 compressibility 319, 408 compressional work 313 Compton effect 208 computer simulations 411 condensate 394 condensation phenomena 383 configurations (microstates) 359 confinement potential (of a quantum dot) 286 conjugate field 404 conservation law 50 conservation theorems 49 conservative forces 28 constraints 48, 89, 91 continuity equation electrodynamics 50, 153 generalizations 156 Minkowski formulation 176 probability current 228 electrodynamics 153, 156 continuous spectrum (absolute continuous) 215 Cooper pairs 277, 395, 396 438 Index Copenhagen interpretation of quantum mechanics 220, 295 Coriolis force 27, 45, 96, 97 Cornu’s spiral 195 correct linear combinations (degenerate perturbation theory) 264 cosmology 4 Coulomb integral 272 Coulomb’s law 8, 110, 119 counter-current principle 323 coupled modes 64 coupled pendula 65 creation (and destruction) operators (harmonic oscillator) 241, 242 critical exponents 330 behavior 332 density 393 droplet radius R c 425 equation of state 332 exponents 331, 407 isotherm 330 quantities 328 region 408 temperature 393, 396, 408 cryotechnology 322 crystal optics 78 crystal optics (birefringence) 188 Curie-Weiss law 330 Curie-Weiss law (ferromagnetism, antiferromagnetism) 304 current density (true versus effective) 165 curvilinear coordinates 126 cut-off frequency ω Debye 401 cyclic coordinates (→ conservation laws) 49, 83, 92 cylindrical coordinates 126, 127 cylindrical symmetry 124 d’Alembert ’s principle 89 equation 165, 166, 168 operator 166 Dalton 335 Davisson and Germer 210 de Broglie 4, 209 de Broglie’s hypothesis of “matter waves” 209, 210, 350 de-icing salt 422 Debye theory (heat capacity of solids) 399 decoupling by diagonalization 64 decoupling of space and time in Galilean transforms 56 degeneracy of the angular momentum (diatomic molecules) 351 degeneracy pressure 389, 391 degenerate Fermi gas 386 perturbation theory 262 electron gas (metals plus white dwarf stars) 390 Fermi gas 418 perturbation theory 262 degrees of freedom 45 demagnetization tensor 142 density operator (statistical operator) 377, 378 derived quantity 7 descending ladder operator 243 determinant 147 deterministic chaos 87 development of stars 387 diagonalization 65 diamagnetism 250, 275 diatomic molecules 340, 341 molecules (rotational energy) 350 molecules (vibrations) 352 dielectric displacement 134 dielectric systems 132, 136 dielectricity (tensorial behavior) 188 difference operation 143 differences in heat capacities (C p − C V , C H − C m ) 318 differential cross-section 42 geometry in a curved Minkowski manifold 96 operation 143 diffraction (Fresnel diffraction versus Fraunhofer diffraction) 193 diffusion of heat 306 dimension 126 dipole limit 133 . mechanics (algebraic methods) 241 accelerated reference frames 95 acceleration (definition) 11 acceleration in planar polar coordinates 31 acceptance (Metropolis algorithm) 412 accuracy for optical. gravitational mass 8 actual (versus virtual) orbits 47, 48 addition rules for angular momenta 255, 270 additivity of partial entropies 415 additivity of the entropy 365, 413, 414 adiabatic expansion. 287, 295 algebraic methods (in quantum mechanics) 241 Alice (quantum cryptography) 289, 297 alkali atoms 256 alternating parity 226 Amp`ere’s law 145 current loops (always equivalent to magnetic