Last change: December 16, 2009

This document is a LaTeX file of 108 pages which contains a lot of equations in physics. It is written at advanced undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and often needs to look up equations.

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Equations in physics - Contents

The table of contents from "Equations in Physics" is shown below.
   Physical Constants                                                 1

1. Mechanics                                                          2
   1.1 Point-kinetics in a fixed coordinate system                    2
       1.1.1 Definitions                                              2
       1.1.2 Polar coordinates                                        2
   1.2 Relative motion                                                2
   1.3 Point-dynamics in a fixed coordinate system                    2
       1.3.1 Force, (angular)momentum and energy                      2
       1.3.2 Conservative force fields                                3
       1.3.3 Gravitation                                              3
       1.3.4 Orbital equations                                        3
             Kepler's equations                                       4
       1.3.5 The virial theorem                                       4
   1.4 Point dynamics in a moving coordinate system                   4
       1.4.1 Apparent forces                                          4
       1.4.2 Tensor notation                                          5
   1.5 Dynamics of masspoint collections                              5
       1.5.1 The centre of mass                                       5
       1.5.2 Collisions                                               6
   1.6 Dynamics of rigid bodies                                       6
       1.6.1 Moment of Inertia                                        6
       1.6.2 Principal axes                                           6
       1.6.3 Time dependence                                          6
   1.7 Variational Calculus, Hamilton and Lagrange mechanics          7
       1.7.1 Variational Calculus                                     7
       1.7.2 Hamilton mechanics                                       7
       1.7.3 Motion around an equilibrium, linearization              7
       1.7.4 Phase space, Liouville's equation                        8
       1.7.5 Generating functions                                     8

2. Electricity & Magnetism                                            9
   2.1 The Maxwell equations                                          9
   2.2 Force and potential                                            9
   2.3 Gauge transformations                                          10
   2.4 Energy of the electromagnetic field                            10
   2.5 Electromagnetic waves                                          10
       2.5.1 Electromagnetic waves in vacuum                          10
       2.5.2 Electromagnetic waves in matter                          11
   2.6 Multipoles                                                     11
   2.7 Electric currents                                              11
   2.8 Depolarizing field                                             12
   2.9 Mixtures of materials                                          12

3. Relativity                                                         13
   3.1 Special relativity                                             13
       3.1.1 The Lorentz transformation                               13
       3.1.2 Red and blue shift                                       14
       3.1.3 The stress-energy tensor and the field tensor            14
   3.2 General relativity                                             14
       3.2.1 Riemannian geometry, the Einstein tensor                 14
       3.2.2 The line element                                         15
       3.2.3 Planetary orbits and the perihelion shift                16
       3.2.4 The trajectory of a photon                               17
       3.2.5 Gravitational waves                                      17
       3.2.6 Cosmology                                                17

4. Oscillations                                                       18
   4.1 Harmonic oscillations                                          18
   4.2 Mechanic oscillations                                          18
   4.3 Electric oscillations                                          19
   4.4 Waves in long conductors                                       19
   4.5 Coupled conductors and transformers                            19
   4.6 Pendulums                                                      19

5. Waves                                                              20
   5.1 The wave equation                                              20
   5.2 Solutions of the wave equation                                 20
       5.2.1 Plane waves                                              20
       5.2.2 Spherical waves                                          21
       5.2.3 Cylindrical waves                                        21
       5.2.4 The general solution in one dimension                    21
   5.3 The stationary phase method                                    21
   5.4 Green functions for the initial-value problem                  22
   5.5 Waveguides and resonating cavities                             22
   5.6 Non-linear wave equations                                      23

6. Optics                                                             24
   6.1 The bending of light                                           24
   6.2 Paraxial geometrical optics                                    24
       6.2.1 Lenses                                                   24
       6.2.2 Mirrors                                                  25
       6.2.3 Principal planes                                         25
       6.2.4 Magnification                                            25
   6.3 Matrix methods                                                 25
   6.4 Aberrations                                                    26
   6.5 Reflection and transmission                                    26
   6.6 Polarization                                                   27
   6.7 Prisms and dispersion                                          27
   6.8 Diffraction                                                    28
   6.9 Special optical effects                                        28
   6.10 The Fabry-Perot interferometer                                29

7. Statistical physics                                                30
   7.1 Degrees of freedom                                             30
   7.2 The energy distribution function                               30
   7.3 Pressure on a wall                                             31
   7.4 The equation of state                                          31
   7.5 Collisions between molecules                                   32
   7.6 Interaction between molecules                                  32

8. Thermodynamics                                                     33
   8.1 Mathematical introduction                                      33
   8.2 Definitions                                                    33
   8.3 Thermal heat capacity                                          33
   8.4 The laws of thermodynamics                                     34
   8.5 State functions and Maxwell relations                          34
   8.6 Processes                                                      35
   8.7 Maximal work                                                   36
   8.8 Phase transitions                                              36
   8.9 Thermodynamic potential                                        37
   8.10 Ideal mixtures                                                37
   8.11 Conditions for equilibrium                                    37
   8.12 Statistical basis for thermodynamics                          38
   8.13 Application to other systems                                  38

9. Transport phenomena                                                39
   9.1 Mathematical introduction                                      39
   9.2 Conservation laws                                              39
   9.3 Bernoulli's equations                                          41
   9.4 Characterising of flows with dimensionless numbers             41
   9.5 Tube flows                                                     42
   9.6 Potential theory                                               42
   9.7 Boundary layers                                                43
       9.7.1 Flow boundary layers                                     43
       9.7.2 Temperature boundary layers                              43
   9.8 Heat conductance                                               44
   9.9 Turbulence                                                     44
   9.10 Self organization                                             44

10. Quantum physics                                                   45
    10.1 Introduction to quantum physics                              45
         10.1.1 Black body radiation                                  45
         10.1.2 The Compton effect                                    45
         10.1.3 Electron diffraction                                  45
    10.2 Wave functions                                               45
    10.3 Operators in quantum physics                                 45
    10.4 The uncertainty principle                                    46
    10.5 The Schr\"odinger equation                                   46
    10.6 Parity                                                       46
    10.7 The tunnel effect                                            47
    10.8 The harmonic oscillator                                      47
    10.9 Angular momentum                                             47
    10.10 Spin                                                        48
    10.11 The Dirac formalism                                         48
    10.12 Atomic physics                                              49
          10.12.1 Solutions                                           49
          10.12.2 Eigenvalue equations                                49
          10.12.3 Spin-orbit interaction                              49
          10.12.4 Selection rules                                     50
    10.13 Interaction with electromagnetic fields                     50
    10.14 Perturbation theory                                         50
          10.14.1 Time-independent perturbation theory                50
          10.14.2 Time-dependent perturbation theory                  51
    10.15 N-particle systems                                          51
         10.15.1 General                                              51
         10.15.2 Molecules                                            52
    10.16 Quantum statistics                                          52

11. Plasma physics                                                    54
    11.1 Introduction                                                 54
    11.2 Transport                                                    54
    11.3 Elastic collisions                                           55
         11.3.1 General                                               55
         11.3.2 The Coulomb interaction                               56
         11.3.3 The induced dipole interaction                        56
         11.3.4 The centre of mass system                             56
         11.3.5 Scattering of light                                   56
    11.4 Thermodynamic equilibrium and reversibility                  57
    11.5 Inelastic collisions                                         57
         11.5.1 Types of collisions                                   57
         11.5.2 Cross sections                                        58
    11.6 Radiation                                                    58
    11.7 The Boltzmann transport equation                             59
    11.8 Collision-radiative models                                   60
    11.9 Waves in plasma's                                            60

12. Solid state physics                                               62
    12.1 Crystal structure                                            62
    12.2 Crystal binding                                              62
    12.3 Crystal vibrations                                           63
         12.3.1 A lattice with one kind of atoms                      63
         12.3.2 A lattice with two kinds of atoms                     63
         12.3.3 Phonons                                               63
         12.3.4 Thermal heat capacity                                 64
    12.4 Magnetic field in the solid state                            65
         12.4.1 Dielectrics                                           65
         12.4.2 Paramagnetism                                         65
         12.4.3 Ferromagnetism                                        65
    12.5 Free electron Fermi gas                                      66
         12.5.1 Thermal heat capacity                                 66
         12.5.2 Electric conductance                                  66
         12.5.3 The Hall-effect                                       66
         12.5.4 Thermal heat conductivity                             67
    12.6 Energy bands                                                 67
    12.7 Semiconductors                                               67
    12.8 Superconductivity                                            68
         12.8.1 Description                                           68
         12.8.2 The Josephson effect                                  69
         12.8.3 Flux quantisation in a superconducting ring           69
         12.8.4 Macroscopic quantum interference                      69
         12.8.5 The London equation                                   70
         12.8.6 The BCS model                                         70

13. Theory of groups                                                  71
    13.1 Introduction                                                 71
         13.1.1 Definition of a group                                 71
         13.1.2 The Cayley table                                      71
         13.1.3 Conjugated elements, subgroups and classes            71
         13.1.4 Isomorfism and homomorfism; representations           72
         13.1.5 Reducible and irreducible representations             72
    13.2 The fundamental orthogonality theorem                        72
         13.2.1 Schur's lemma                                         72
         13.2.2 The fundamental orthogonality theorem                 72
         13.2.3 Character                                             72
    13.3 The relation with quantum mechanics                          73
         13.3.1 Representations, energy levels and degeneracy         73
         13.3.2 Breaking of degeneracy with a perturbation            73
         13.3.3 The construction of a basefunction                    73
         13.3.4 The direct product of representations                 74
         13.3.5 Clebsch-Gordan coefficients                           74
         13.3.6 Symmetric transformations of operators,
                irreducible tensor operators                          74
         13.3.7 The Wigner-Eckart theorem                             75
    13.4 Continuous groups                                            75
      13.4.1 The 3-dimensional translation group                      75
      13.4.2 The 3-dimensional rotation group                         76
      13.4.3 Properties of continuous groups                          76
    13.5 The group SO(3)                                              77
    13.6 Applications to quantum mechanics                            78
         13.6.1 Vectormodel for the addition of angular momentum      78
         13.6.2 Irreducible tensoroperators, matrixelements and
                selection rules                                       78
                Some examples of the behaviour of operators
                under SO(3)                                           78
                Selection rules for dipole transitions                79
                Land\'e -equation for the anomalous
                Zeeman splitting                                      79
    13.7 Applications to particle physics                             79

14. Nuclear physics                                                   81
    14.1 Nuclear forces                                               81
    14.2 The shape of the nucleus                                     82
    14.3 Radioactive decay                                            82
    14.4 Scattering and nuclear reactions                             83
         14.4.1 Kinetic model                                         83
         14.4.2 Quantum mechanical model for n-p scattering           83
         14.4.3 Conservation of energy and momentum in nuclear
                reactions                                             84
    14.5 Radiation dosimetry                                          84

15. Quantum field theory & Particle physics                           85
     15.1 Creation and annihilation operators                         85
     15.2 Classical and quantum fields                                85
     15.3 The interaction picture                                     86
     15.4 Real scalar field in the interaction picture                86
     15.5 Charged spin-0 particles, conservation of charge            87
     15.6 Field functions for 1/2-particles                           87
     15.7 Quantization of spin-1/2 fields                             88
     15.8 Quantization of the electromagnetic field                   89
     15.9 Interacting fields and the S-matrix                         89
     15.10 Divergences and renormalization                            90
     15.11 Classification of elementary particles                     90
     15.12 P and CP-violation                                         92
     15.13 The standard model                                         93
           15.13.1 The electroweak theory                             93
           15.13.2 Spontaneous symmetry breaking: the Higgs mechanism 94
           15.13.3 Quantumchromodynamics                              94
    15.14 Pathintegrals                                               95
    15.15 Unification and quantum gravity                             95

16. Astrophysics                                                      96
    16.1 Determination of distances                                   96
    16.2 Brightness and magnitudes                                    96
    16.3 Radiation and stellar atmospheres                            97
    16.4 Composition and evolution of stars                           97
    16.5 Energy production in stars                                   98

The Nabla operator                                                    99
The SI units                                                         100