Some
Semiquantitative
Considerations
on
the
Development
of
Stars
387
53.3
Bose-Einstein
Condensation
391
53.4
Ginzburg-Landau
Theory
of
Superconductivity
395
53.5
Debye
Theory
of
the
Heat
Capacity
of
Solids
399
53.6
Landau's
Theory
of
2nd-order
Phase
Transitions
403
53.7
Molecular
Field
Theories;
Mean
Field
Approaches
405
53.8
Fluctuations
408
53.9
Monte
Carlo
Simulations
411
54
Applications
II:
Phase
Equilibria
in
Chemical
Physics
413
54.1
Additivity
of
the
Entropy;
Partial
Pressure;
Entropy
of
Mixing
413
54.2
Chemical
Reactions;
the
Law
of
Mass
Action
416
54.3
Electron
Equilibrium
in
Neutron
Stars
417
54.4
Gibbs'
Phase
Rule
419
54.5
Osmotic
Pressure
420
54.6
Decrease
of
the
Melting
Temperature
Due
to
"De-icing"
Salt.
422
54.7
The
Vapor
Pressure
of
Spherical
Droplets
423
55
Conclusion
to
Part
IV
427
References
431
Index
435
Vorwort
Preface
This textbook on theoretical physics (I-IV) is based on lectures held by one of the authors at the University of Regensburg in Germany. The four 'canonical' parts of the subject have been condensed here into a single volume with the following main sections :
I - Mechanics and Basic Relativity;
II - Electrodynamics and Aspects of Optics;
III - Quantum Mechanics (non-relativistic theory), and
IV - Thermodynamics and Statistical Physics.
Our compendium is intended primarily for revision purposes and/or to aid in a deeper understanding of the subject. For an introduction to theoretical physics many standard series of textbooks, often containing seven or more volumes, are already available (see, for example, [1]).
Exercises closely adapted to the book can be found on one of the authors websites [2], and these may be an additional help.
We have laid emphasis on relativity and other contributions by Einstein, since the year 2005 commemorated the centenary of three of his groundbreaking theories.
In Part II (Electrodynamics) we have also treated some aspects with which every physics student should be familiar, but which are usually neglected in textbooks, e.g., the principles behind cellular (or mobile) phone technology, synchrotron radiation and holography. Similarly, Part III (Quantum Mechanics) additionally covers aspects of quantum computing and quantum cryptography.
We have been economical with figures and often stimulate the reader to sketch his or her own diagrams. The frequent use of italics and quotation marks throughout the text is to indicate to the reader where a term is used in a specialized way. The Index contains useful keywords for ease of reference.
Finally we are indebted to the students and colleagues who have read parts of the manuscript and to our respective wives for their considerable support.
Regensburg, May 2007
Uwe Krey
Anthony Owen
Klappentext
Uwe Krey
Anthony Owen
Basic Theoretical Physics
This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I. Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers' insight and add new dimensions to their understanding of these fundamental concepts. Recent topics such as holography and quantum cryptography are included, thus making this a unique contribution to the learning material for theoretical physics.
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 (Cp - CV, CH - Cm) 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 - moment 133 - representation (of magnetic bodies) 140 - fields 132 Dirac - 's abstract bra-ket formalism 215 - 's abstract quantum mechanics 210 - 's delta function 112, 133, 138, 214 - 's equivalent spin operator (Heitler-London theory) 273 - 's relativistic quantum theory (not described) 251 - representation (interaction) 245 Dirichlet's problem 127, 128 discrete-line radiation 179 dispersion - phenomena 346 - relation ω {k) 179, 399, 402 double pendulum 85, 87 droplet radius: critical value 425 Drude theory (of the electric resistance) 164 dual space 78 duality (wave-particle) 208 Dulong-Petit's law 402 dumbbell approximation (for a dipole)132, 140 dynamic conserved quantities 50 dynamic susceptibility 410 Dyson series (interaction representation) 247 Dyson's time-ordering operator 247
- 's theory of gravity 33 - and de Haas 151 - hidden variables 279 Einstein, Podolski, and Rosen 279, 290, 291, 295 Einstein, Podolski, and Rosen's "acceptability" postulates (reality, locality, retardation) 280 electric - dipole radiation 173 - octupole radiation 173 - polarization P 133, 134, 139 electro-mechanical equivalent of heat 306 electromagnetic waves 163 - waves (planar, spherical) 166 energy theorem 163 electromotive force 153 electron equilibrium in neutron stars 417 electrons in metals (Sommerfeld formalism) 383 electrostatics 119 elevator force 96 ellipsoid of revolution 77, 190 emission (induced versus spontaneous) 269 Emperor's bell in Cologne 75 encoding (decoding) of a computer message 284 energy - momentum four-vector 103 - dissipation 164 - conservation 14 - conservation for the electromagnetic field 163 - density of the electric field 142 - density of the magnetic field 142 - theorem (electromagnetic) 163 - theorem (mechanics) 27, 28 - conservation 50 entanglement 279, 291, 295 enthalpy I{T, p, N) 319, 320, 428 entropy - S(T, V, N) 301, 315, 350, 357, 427, 429 - constant 415 - from the microcanonical ensemble 363 - of mixing 413, 414, 416, 421 equations of motion - Hamilton 53 - Lagrange 53 - Newton 5, 53 - Lagrange 45 equipartition of energy 352 equivalence - of inertial frames 58 - of mass and energy (Einstein) 104 - principle (Einstein) 96 ergodic systems 367, 411, 412 Euler - 's relation (Moivre) 159 - Lagrange equations 48 - angles φ , υ , ψ 81, 82, 91 equations of a rigid body 79 even normal modes 67 even parity 67 exchange integral 273 expansion theorem 215 expectation value in quantum mechanics 218 extensive versus intensive variables (thermodynamics) 314 extensive versus intensive work 320, 354 external force 23 external torque 24 extraordinary versus ordinary beam (uniaxial birefringence) 191 extremal paths (Einstein) 96
far-field range 170, 173 Faraday's induction law 124, 153, 155 Fermat 's variational principle of the shortest optical path 55, 185, 187 Fermi - 's "golden rules" 179, 267, 269 - and Bose gases 335, 337, 379 - and Bose statistics 337 - energy eF 383, 394 - liquid 383 fermions 252 fermions and bosons 379, 383
ferroelectric systems 136
ferromagnetism - = interplay of Coulomb interaction - and Pauli's principle 274 - ideal 304 - in solids 275 Feynman diagram for induced absorption 269 Fick's laws (heat diffusion) 306, 307 fictitious (=inertial) forces 95 field - energy conservation 163 - energy 141 - tensor (skew-symmetric) in electromagnetism 176
figure axis 84
fine structure (of the spectra) 251, 256, 294 First and Second Laws of Thermodynamics: Equivalent Formulations 315 First Law of Thermodynamics 313
first-order phase transitions 403 fixed cartesian vectors for a rigid body 79
fluctuation-dissipation theorem 410
fluctuations 332, 333, 408, 428
fluid (liquid or gas) 305
flux tubes (vortex lattice) 398 Fock-Darwin theory 287 force - power four-vector 102 - and torque on electric and magnetic dipoles 25, 139 - of constraint 13, 90, 91 - and torque on electric and magnetic dipoles 140 Foucault pendulum 98
fountain effect (superfluidity) 394
four-vectors (Minkowski) 58 Fourier methods (heat conduction) 308 Fourier transformation (Fraunhofer diffraction) 197 Fraunhofer diffraction at rectangular - and circular apertures 197 free - energy (Gibbs) Fg (T, V, H, N) 366 - energy (Helmholtz) F(T, V, N) 301, 353, 366, 396, 397, 427 - energy (Helmholtz) of droplets 423 - energy for Landau's theory 404 - enthalpy per particle (= μ in fluids) 365, 416, 418
frequency dispersion 179 Fresnel - 's integrals 195 - ellipsoid (E-ellipsoid) 189 - diffraction at an edge 193 - diffraction versus Fraunhofer diffraction 193
frictional force 17, 27, 45
frictional heat 305
fringes (in Fresnel's diffraction from an edge) 196
from reversibility (quantum mechanics) to irreversibility: Fermi's "golden rules" 268
ladder operators (ascending or descending) 243 Lagrange formalism 89 Lagrange multiplier 90 Lagrangian - equations of the 1st kind 48, 91 - equations of the 2nd kind 45, 48 - equations of the 2nd kind with - constraints 91
- for the Lorentz force 60
- function 45
- equations of the 2nd kind 47 Lande factor gj(L, S) 265, 406 Landau's theory (second-order phase transitions) 403 Laplace identity 72 laser 199 laser cooling (ultralow temperatures) 371 latent heat (e.g., vaporization and melting) 369 latitude υ 126 lattice-gas interpretation (Ising model) 332, 333, 425 law of mass action (chemical reactions)417 least action (Hamilton's principle) 55 left-handedness 71 Legendre polynomial 239 Legendre transform 428 Legget, Abrikosov, Ginzburg 395 Lenard (photoeffect) 208 length contraction 59 Lennard-Jones potential 323, 327 Lenz vector 39, 50, 236 light quanta (photons) 379 Linde process (liquefaction) 322 linear approximation (mean field) of a Hamiltonian 334 linearity (e.g., of Maxwell's equations) 120 linearity (of a theory) 21 liquefaction (Linde) 319, 322, 323 local inertial frames (Einstein) 5, 96 locality in quantum mechanics 279 locally orthogonal curvilinear coordinates 126 longitude φ 126 Lorentz - force 27, 45, 153, 155, 217, 295 - force (mks-A versus cgs) 112 - gauge 165, 176 - transformations 56 - transformations for the electromagnetic fields 177 - force 60 - gauge 166 - transformations 175 Loschmidt's number LQ (the number of molecules in a mole) 303 losses of a response process 410
n-th order phase transitions (Ehrenfest) 403 Nabla operator 114 near-field microscopy 194, 197 near-field range 170 Nernst's heat theorem (Third Law)316, 372, 374 Neumann's problem 129, 130 neutrino 418 neutron 418 neutron stars 97, 387, 389, 418 Newton - 's 3rd axiom, weak and strong forms 8
- 's bucket experiment 98
- 's equations of motion 5, 293
- 's particle interpretation of light184
- 's third axiom 26, 38, 119 Newtonian - mechanics 174 - synthesis of Kepler's laws 33 - analysis of Kepler's laws 33, 36, 39 nodal theorem 226, 240 Noether's theorem 49, 50 non-additivity of the partial pressures for real gases 414 non-autonomous 86 non-degenerate perturbation theory 262 non-existence of a local maximum of a harmonic function 129 non-integrable systems 85 non-linearity (e.g. of the chromodynamic equations for nuclear physics) 120 non-locality of quantum mechanics (singlet decay) 281, 295 non-perturbative terms 263 non-reality of quantum mechanics (states are "prepared", not "stated") 281 normal fluid 394 - frequencies (eigenfrequencies) 63 - modes 63 north anomaly 100 nuclear transitions 180 nucleation processes 425 nuclei 180 number operator 242 numerical methods (electric fields) 131 nutation 84 Nyquist theorem 411
- 's reduced constant h 343 - general remarks (if at all) 4, 207, 361 planetary motion 31 plasmons (quanta of charge-density waves) 185 Poincaré - 's lemma 125, 147 - repetition time 268 - section 86 Poinsot - 's construction (for rigid bodies) 77, 189
- 's construction (in crystal optics)189, 191 plane 78, 80 point charge 114 point mass 5, 23 Poisson brackets 50, 53 Poisson's equation 128 polar (versus axial) vectors 71 polarizable matter 132 polarization (electric) 133 - representation (versus effectivecharge representation) 137 - contributions (to the effective densities of charge and current) 165 polarons 379 Popper (philosophy of science) 3 position representation ("wave mechanics") 231 position representation (versus momentum representation or matrix representations) 211 potential energy 14, 26-28, 46 power (relativistic) 102 power balance (dissipative versus reactive parts) 162 Poynting vector 163, 167, 172 pre-Einstein (aether) 58 precession (of a top) 84 principal moments of inertia 77 principal quantum number n (shell index) 237 principle - of "maximal proper time" (Einstein) 60 - of d'Alembert 89 - of relativity (Einstein) 58 - of superposition 18, 33 - probability current 228 probability statements (for the results of measurements in quantum mechanics) 294 profile function (of a traveling wave) 167 projectile and target 41 proper time (eigenzeit) 55, 59 proton 418 pseudo-length 59 pseudo-perpendicular 102 pseudo-rotation of the Minkowski space (Lorentz transformation) 59, 101, 175 pseudo-scalar product 59 pulsar (neutron star) 391
radiation from an oscillating electric dipole 170 radiation from an oscillating magnetic dipole 171 radius of curvature of a line in space 12 ray optics (geometrical optics) 185 ray optics versus wave optics 185 Rayleigh-Jeans' law (black-body radiation) 207, 345 rays in a complex Hilbert space (equivalence classes of Hilbert vectors distinguished by a complex factor) 211 Real Gases (as opposed to Ideal Gases) 304 reality in quantum mechanics 279 reconstruction wave (in holography) 200 red giant (stars) 389 reduced - length of a physical pendulum 75 - variables (harmonic oscillator) 231 - variables (hydrogen atom) 237 reference frames 95 reference wave 199 reflectivity R and transmittivity T at steps in the potential energy 227 reflexion (and transmission) 226 rejection (Metropolis algorithm) 412 relative motion 29 relative permeability 142 relativistic invariance of Maxwell's theory 155, 174 relativistic quantum field theories (like QED or QCD) 4 relaxation frequencies 19 relaxation time 17 repetition time (Poincaré) 268 representation - (change of ...) 245
- (= "aspect") 245
- (Schrödinger, Heisenberg, Dirac) 246 - in terms of (effective) magnetic - charges 140 reproduction of wave functions with spin by doubled rotations 252 resolution of the identity (spectral resolution) 215 resonance 21 resonance (in alternating-current circuits) 158, 161 response 410 rest energy (relativistic) 103 rest mass 7 retardation between cause and effect 169 retarded potentials 165, 169 reversible processes 315 reversion pendulum 76 rheonomous 45, 46, 89 Riemann zeta function 393 right-handed rectangular trihedra (in radiation fields) 171 right-handedness 71, 149 rigid bodies 71 Ritz minimization 259 roller on an inclined plane 74, 91 rosette orbits 35, 39 rotation of wave functions with spin 251 rotational - energy (diatomic molecules) 350 - energy (rigid body) 71, 73 part of the kinetic energy of a molecule 340 Runge-Lenz vector 39, 50, 236 Rutherford 209 Rutherford scattering cross-section 41, 42 Rydberg 209, 237
target (in scattering phenomena) 41 technical friction factor 15 temperature 303 test functions 113, 214 test message (quantum cryptography) 297 thermal - fluctuations 332 - (versus caloric) equation of state 317 - coherence length ξ (T) 405 - equilibrium 306 thermodynamic temperature T (Kelvin) 301 thermostat 337 Third Law of Thermodynamics 316, 371, 372, 429 Thomson: impossibility of an ideal heat machine 355 throttle valve (Joule-Thomson) 321 time dilatation 59 time-ordering operator (Dyson) 247 torque on electric dipoles 25 torque-free top 78 total Pfaffian form 29 trace formalism 377 trade winds (Passat) 98 transfer matrix method 228 transformation between mksa and cgs units 111 transformer 158, 159 transient time 169 transition to chaos 80 translational and rotational parts of the kinetic energy 71 translational part of the kinetic energy 337, 339 transversality (of electromagnetic waves) 167 traps (ultralow temperatures) 371 traveling wave 167 triangulation 116 triple point 305, 419 triplet pairing (in He3) 278 triplet wave functions (two spins) 271, 272, 276 true charges (as opposed to effective ones) 137 true force (as opposed to inertial forces) 95 tunneling states (two levels) 402 through a barrier 229 transmittivity 229 two-level systems in quantum computing 283 two-particle problems 29 two-spinors 249 type II superconductors (vortex lattice) 398
ultralow temperatures 371, 374 ultrarelativistic behavior and photon gases 338 ultraviolet (spectral range) 179 unattainability of T = 0 in a finite number of steps 372, 373 undulator 173 uniaxial crystals (normal birefringence) 190 uniqueness of the solution for a capacitor problem 129 unitarity relation 226, 227 unitary - equivalence 245 - operators 245, 283 - vector space (pre-Hilbert space) 212 - operators 53 units in electromagnet ism 109 units of length and time 6
vacuum light velocity 169 vacuum permeability 111, 250 van der Waals equation of state 304, 317, 322, 330, 425 van der Waals theory 327, 369 vapor pressure outside a droplet 423 variational - derivative 48 - parameter 47 - principle of "least action" - principle of "maximal proper time" (Einstein) 60 vector model (quantization of the angular-moment urn) 257 vector potential 60, 147 velocity - definition 11 - of light in vacuo 111 - transformation (non-relativistic versus relativistic) 58 vibrational part of the energy of a molecule 340 vibrations of diatomic molecules 352 virtual displacement 89 virtual orbits 47-49 vortices and sources of a field 146
Zeeman effect (external magnetic field) 250, 264 Zeeman term (in the Hamiltonian)354 zero-point pressure ("degeneracy pressure") 387
Autor
Prof. Dr. Uwe Krey
University of Regensburg (retired)
FB Physik
Universitätsstraße 31
93053 Regensburg, Germany
E-mail: uwe.krey@physik.uni-regensburg.de
Dr. rer nat habil Anthony Owen
University of Regensburg (retired)
FB Physik
Universitätsstraße 31
93053 Regensburg, Germany
E-mail: anthony.owen@physik.uni-regensburg.de
Reviews
From the reviews: "A comprehensive work covering the material that graduate students in physics typically would study in preparing for doctoral candidacy examinations. ... This book would be very useful for self-study by motivated students, or for preparation for candidacy exams. ... Practicing physicists may find that the brief, accessible treatments of many topics will earn this book a place on a convenient bookshelf. Summing Up: Recommended. Upper-division undergraduates through professionals." (M. C. Ogilvie, CHOICE, Vol. 45 (7), 2008) "The book, written by two ... 'working physicists', contains what the authors regard as being 'basic knowledge' in the standard courses of theoretical physics (yet) held at German Universities. ... is primarily intended to cover the 'Basic Theoretical Physics' in a single and handy volume. ... Hence, the book should be considered as being a kind of 'compendium' of ... formulas used in theoretical physics where the formulas are filled in between with some remarks." (Jürgen Tolksdorf, Zentralblatt MATH, Vol. 1134 (12), 2008)
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