Basic Theoretical Physics: A Concise Overview P2 pptx

XII Contents 28.3 Unitary Equivalence; Change of Representation . . . . . . . . . . . . 245 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem) 249 29.1 Spin Momentum; the Hamilton Operator with Spin-orbit Interaction. . . . . . . . . . 249 29.2 Rotation of Wave Functions with Spin; Pauli’s Exclusion Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 30 Addition of Angular Momenta 255 30.1 Composition Rules for Angular Momenta . . . . . . . . . . . . . . . . . . 255 30.2 Fine Structure of the p-Levels; Hyperfine Structure . . . . . . . . . 256 30.3 Vector Model of the Quantization of the Angular Momentum 257 31 Ritz Minimization 259 32 Perturbation Theory for Static Problems 261 32.1 FormalismandResults 261 32.2 Application I: Atoms in an Electric Field; The Stark Effect . . 263 32.3 Application II: Atoms in a Magnetic Field; Zeeman Effect . . . 264 33 Time-dependent Perturbations 267 33.1 Formalism and Results; Fermi’s “Golden Rules” . . . . . . . . . . . . 267 33.2 SelectionRules 269 34 Magnetism: An Essentially Quantum Mechanical Phenomenon 271 34.1 Heitler and London’s Theory of the H 2 -Molecule 271 34.2 Hund’s Rule. Why is the O 2 -Molecule Paramagnetic? . . . . . . . 275 35 Cooper Pairs; Superconductors and Superfluids 277 36 On the Interpretation of Quantum Mechanics (Reality?, Locality?, Retardation?) 279 36.1 Einstein-Podolski-RosenExperiments 279 36.2 The Aharonov-Bohm Effect; Berry Phases . . . . . . . . . . . . . . . . . 281 36.3 QuantumComputing 283 36.4 2dQuantumDots 285 36.5 Interaction-free Quantum Measurement; “WhichPath?”Experiments 287 36.6 Quantum Cryptography 289 37 Quantum Mechanics: Retrospect and Prospect 293 38 Appendix: “Mutual Preparation Algorithm” for Quantum Cryptography 297 Contents XIII Part IV Thermodynamics and Statistical Physics 39 Introduction and Overview to Part IV 301 40 Phenomenological Thermodynamics: Temperature and Heat 303 40.1 Temperature 303 40.2 Heat 305 40.3 Thermal Equilibrium and Diffusion of Heat . . . . . . . . . . . . . . . . 306 40.4 Solutionsofthe DiffusionEquation 307 41 The First and Second Laws of Thermodynamics 313 41.1 Introduction:Work 313 41.2 First and Second Laws: Equivalent Formulations . . . . . . . . . . . 315 41.3 Some Typical Applications: C V and ∂U ∂V ; TheMaxwellRelation 316 41.4 GeneralMaxwellRelations 318 41.5 The Heat Capacity Differences C p − C V and C H − C m 318 41.6 Enthalpy and the Joule-Thomson Experiment; LiquefactionofAir 319 41.7 Adiabatic ExpansionofanIdealGas 324 42 Phase Changes, van der Waals Theory and Related Topics 327 42.1 VanderWaals Theory 327 42.2 Magnetic Phase Changes; The Arrott Equation. . . . . . . . . . . . . 330 42.3 Critical Behavior; Ising Model; Magnetism and Lattice Gas . . 332 43 The Kinetic Theory of Gases 335 43.1 Aim 335 43.2 TheGeneralBernoulliPressureFormula 335 43.3 Formulafor PressureinanInteractingSystem 341 44 Statistical Physics 343 44.1 Introduction; Boltzmann-Gibbs Probabilities . . . . . . . . . . . . . . . 343 44.2 The Harmonic Oscillator and Planck’s Formula . . . . . . . . . . . . . 344 45 The Transition to Classical Statistical Physics 349 45.1 The Integral over Phase Space; Identical Particles in Classical StatisticalPhysics 349 45.2 The Rotational Energy of a Diatomic Molecule . . . . . . . . . . . . . 350 XIV Contents 46 Advanced Discussion of the Second Law 353 46.1 FreeEnergy 353 46.2 On the Impossibility of Perpetual Motion oftheSecond Kind 354 47 Shannon’s Information Entropy 359 48 Canonical Ensembles in Phenomenological Thermodynamics 363 48.1 ClosedSystemsandMicrocanonicalEnsembles 363 48.2 The Entropy of an Ideal Gas fromtheMicrocanonicalEnsemble 363 48.3 Systems in a Heat Bath: Canonical and Grand Canonical Distributions . . . . . . . . . . . . . . 366 48.4 From Microcanonical to Canonical and Grand Canonical Ensembles 367 49 The Clausius-Clapeyron Equation 369 50 Production of Low and Ultralow Temperatures; Third Law 371 51 General Statistical Physics (Statistical Operator; Trace Formalism) 377 52 Ideal Bose and Fermi Gases 379 53 Applications I: Fermions, Bosons, Condensation Phenomena 383 53.1 ElectronsinMetals (SommerfeldFormalism) 383 53.2 Some Semiquantitative Considerations on the Development ofStars 387 53.3 Bose-EinsteinCondensation 391 53.4 Ginzburg-Landau Theory of Superconductivity . . . . . . . . . . . . . 395 53.5 Debye Theoryofthe HeatCapacityofSolids 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 MonteCarloSimulations 411 54 Applications II: Phase Equilibria in Chemical Physics 413 54.1 Additivity of the Entropy; Partial Pressure; EntropyofMixing 413 54.2 ChemicalReactions;the LawofMassAction 416 54.3 Electron Equilibrium in Neutron Stars . . . . . . . . . . . . . . . . . . . . 417 54.4 Gibbs’PhaseRule 419 Contents XV 54.5 OsmoticPressure 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 Part I Mechanics and Basic Relativity 1 Space and Time 1.1 Preliminaries to Part I This book begins in an elementary way, before progressing to the topic of an- alytical mechanics. 1 Nonlinear phenomena such as “chaos” are treated briefly in a separate chapter (Chap. 12). As far as possible, only elementary formulae have been used in the presentation of relativity. 1.2 General Remarks on Space and Time a) Physics is based on experience and experiment, from which axioms or gen- erally accepted principles or laws of nature are developed. However, an axiomatic approach, used for the purposes of reasoning in order to estab- lish a formal deductive system, is potentially dangerous and inadequate, since axioms do not constitute a necessary truth, experimentally. b) Most theories are only approximate, preliminary, and limited in scope. Furthermore, they cannot be proved rigorously in every circumstance (i.e., verified), only shown to be untrue in certain circumstances (i.e., falsified; Popper). 2 For example, it transpires that Newtonian mechanics only ap- pliesaslongasthemagnitudesofthevelocitiesoftheobjectsconsidered are very small compared to the velocity c of light in vacuo. c) Theoretical physics develops (and continues to develop) in “phases” (Kuhn 3 , changes of paradigm). The following list gives examples. 1. From ∼ 1680−1860: classical Newtonian mechanics,falsifiedbyexper- iments of those such as Michelson and Morley (1887). This falsification was ground-breaking since it led Einstein in 1905 to the insight that the perceptions of space and time, which were the basis of Newtonian theory, had to be modified. 2. From ∼ 1860−1900: electrodynamics (Maxwell). The full consequences of Maxwell’s theory were only later understood by Einstein through his special theory of relativity (1905), which concerns both Newtonian 1 See, for example, [3]. 2 Here we recommend an internet search for Karl Popper. 3 For more information we suggest an internet search for Thomas Samuel Kuhn. 4 1 Space and Time mechanics (Part I) and Maxwell’s electrodynamics (Part II). In the same year, through his hypothesis of quanta of electromagnetic waves (photons), Einstein also contributed fundamentally to the developing field of quantum mechanics (Part III). 3. 1905: Einstein’s special theory of relativity, and 1916: his general theory of relativity. 4. From 1900: Planck, Bohr, Heisenberg, de Broglie, Schr¨odinger: quan- tum mechanics; atomic and molecular physics. 5. From ∼1945: relativistic quantum field theories, quantum electro- dynamics, quantum chromodynamics, nuclear and particle physics. 6. From ∼1980: geometry (spacetime) and cosmology: supersymmetric theories, so-called ‘string’ and ‘brane’ theories; astrophysics; strange matter. 7. From ∼1980: complex systems and chaos; nonlinear phenomena in mechanics related to quantum mechanics; cooperative phenomena. Theoretical physics is thus a discipline which is open to change. Even in mechanics, which is apparently old-fashioned, there are many unsolved prob- lems. 1.3 Space and Time in Classical Mechanics Within classical mechanics it is implicitly assumed – from relatively inaccu- rate measurements based on everyday experience – that a) physics takes place in a three-dimensional Euclidean space that is not influenced by material properties and physical events. It is also assumed that b) time runs separately as an absolute quantity; i.e., it is assumed that all clocks can be synchronized by a signal transmitted at a speed v →∞. Again, the underlying experiences are only approximate, e.g., that α) measurements of lengths and angles can be performed by translation and rotation of rigid bodies such as rods or yardsticks; β) the sum of the interior angles of a triangle is 180 ◦ , as Gauss showed in his famous geodesic triangulation of 1831. Thus, according to the laws of classical mechanics, rays of light travel in straight lines (rectilinear behavior). Einstein’s prediction that, instead, light could travel in curved paths became evident as a result of very accurate as- tronomical measurements when in 1919 during a solar eclipse rays of light traveling near the surface of the sun were observed showing that stellar bod- ies under the influence of gravitation give rise to a curvature of spacetime (general theory of relativity), a phenomenon which was not measurable in Gauss’s time. Assumption b) was also shown to be incorrect by Einstein (see below). 2 Force and Mass 2.1 Galileo’s Principle (Newton’s First Axiom) Galileo’s principle, which forms the starting point of theoretical mechanics, states that in an inertial frame of reference all bodies not acted upon by any force move rectilinearly and homogeneously at constant velocity v. The main difficulty arising here lies in the realization of an inertial frame, which is only possible by iteration: to a zeroth degree of approximation an inertial frame is a system of Cartesian coordinates, which is rigidly rotating with the surface of the earth, to which its axes are attached; to the next approximation they are attached to the center of the earth; in the following approximation they are attached to the center of the sun, to a third approxi- mation to the center of our galaxy, and so on. According to Mach an inertial frame can thus only be defined by the overall distribution of the stars. The final difficulties were only resolved later by Einstein, who proposed that in- ertial frames can only be defined locally, since gravitation and acceleration are equivalent quantities (see Chap. 14). Galileo’s principle is essentially equivalent to Newton’s First Axiom (or Newton’s First Law of Motion). 2.2 Newton’s Second Axiom: Inertia; Newton’s Equation of Motion This axiom constitutes an essential widening and accentuation of Galileo’s principle through the introduction of the notions of force, F ,andinertial mass, m t ≡ m. (This is the inertial aspect of the central notion of mass, m.) Newton’s second law was originally stated in terms of momentum. The rate of change of momentum of a body is proportional to the force acting on the body and is in the same direction. where the momentum of a body of inertial mass m t is quantified by the vector p := m t ·v. 1 Thus F = dp dt . (2.1) 1 Here we consider only bodies with infinitesimal volume: so-called point masses. 62ForceandMass The notion of mass also has a gravitational aspect, m s (see below), where m t = m s (≡ m). However, primarily a body possesses ‘inertial’ mass m t , which is a quantitative measure of its inertia or resistance to being moved 2 . (Note: In the above form, (2.1) also holds in the special theory of relativity, see Sect. 15 below, according to which the momentum is given by p = m 0 v  1 − v 2 c 2 ; m 0 is the rest mass, which only agrees with m t in the Newtonian approxi- mation v 2  c 2 ,wherec is the velocity of light in vacuo.) Equation (2.1) can be considered to be essentially a definition of force involving (inertial) mass and velocity, or equivalently a definition of mass in terms of force (see below). As already mentioned, a body with (inertial) mass also produces a gravi- tational force proportional to its gravitational mass m s . Astonishingly, in the conventional units, i.e., apart from a universal constant, one has the well-known identity m s ≡ m t , which becomes still more astonishing, if one simply changes the name and thinks of m s as a “gravitational charge” instead of “gravitational mass”. This remarkable identity, to which we shall return later, provided Einstein with strong motivation for developing his general theory of relativity. 2.3 Basic and Derived Quantities; Gravitational Force The basic quantities underlying all physical measurements of motion are – time: defined from multiples of the period of a so-called ‘atomic clock’, and – distance: measurements of which are nowadays performed using radar signals. The conventional units of time (e.g., second, hour, year) and length (e.g., kilometre, mile, etc.) are arbitrary. They have been introduced historically, often from astronomical observations, and can easily be transformed from one to the other. In this context, the so-called “archive metre” (in French: “m`etre des archives”) was adopted historically as the universal prototype for a standard length or distance: 1 metre (1 m). Similarly, the “archive kilogram” or international prototype kilogram in Paris is the universal standard for the unit of mass: 1 kilogram (1 kg). 2 in German: inertial mass = tr¨age Masse as opposed to gravitational mass = schwere Masse m s . The fact that in principle one should distinguish between the two quantities was already noted by the German physicist H. Hertz in 1884; see [4]. 2.3 Basic and Derived Quantities; Gravitational Force 7 However, the problem as to whether the archive kilogram should be used as a definition of (inertial) mass or a definition of force produced a dilemma. In the nineteen-fifties the “kilopond (kp)” (or kilogram-force (kgf)) was adopted as a standard quantity in many countries. This quantity is defined as the gravitational force acting on a 1 kg mass in standard earth gravity (in Paris where the archive kilogram was deposited). At that time the quantity force was considered to be a “basic” quantity, while mass was (only) a “de- rived” one. More recently, even the above countries have reverted to using length, time,and(inertial) mass as base quantities and force as a derived quantity. In this book we shall generally use the international system (SI) of units, which has 7 dimensionally independent base units: metre, kilogram, second, ampere, kelvin, mole and candela. All other physical units can be derived from these base units. What can be learnt from this? Whether a quantity is basic or (only) de- rived,isamatter of convention.Eventhenumber of base quantities is not fixed; e.g., some physicists use the ‘cgs’ system, which has three base quan- tities, length in centimetres (cm), time in seconds (s) and (inertial) mass in grams (g), or multiples thereof; or the mksA system, which has four base quantities, corresponding to the standard units: metre (m), kilogram (kg), second (s) and ampere (A) (which only comes into play in electrodynamics). Finally one may adopt a system with only one basic quantity, as preferred by high-energy physicists, who like to express everything in terms of a funda- mental unit of energy, the electron-volt eV: e.g., lengths are expressed in units of  ·c/(eV), where  is Planck’s constant divided by 2π, which is a universal quantity with the physical dimension action = energy ×time, while c is the velocity of light in vacuo; masses are expressed in units of eV/c 2 ,whichisthe “rest mass” corresponding to the energy 1 eV. (Powers of  and c are usually replaced by unity 3 ). As a consequence, writing Newton’s equation of motion in the form m · a = F (2.2) (relating acceleration a := d 2 r dt 2 and force F ), it follows that one can equally well say that in this equation the force (e.g., calibrated by a certain spring) is the ‘basic’ quantity, as opposed to the different viewpoint that the mass is ‘basic’ with the force being a derived quantity, which is ‘derived’ by the above equation. (This arbitariness or dichotomy of viewpoints reminds us of the question: “Which came first, the chicken or the egg?!”). In a more modern didactical framework based on current densities one could, for example, write the left-hand side of (2.2) as the time-derivative of the momentum, dp dt ≡ F , thereby using the force as a secondary quantity. However, as already 3 One should avoid using the semantically different formulation “set to 1” for the quantities with non-vanishing physical dimension such as c(= 2.998 · 10 8 m/s), etc. . Experiment; LiquefactionofAir 319 41.7 Adiabatic ExpansionofanIdealGas 324 42 Phase Changes, van der Waals Theory and Related Topics 327 42.1 VanderWaals Theory 327 42.2 Magnetic Phase Changes; The Arrott Equation begins in an elementary way, before progressing to the topic of an- alytical mechanics. 1 Nonlinear phenomena such as “chaos” are treated briefly in a separate chapter (Chap. 12). As far as possible,. events. It is also assumed that b) time runs separately as an absolute quantity; i.e., it is assumed that all clocks can be synchronized by a signal transmitted at a speed v →∞. Again, the underlying