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.

A list of recent additions and improvements is available.

1. physics.tex: an English LaTeX file (Approximately 310kb).
2. physics.zip: a zipped English LaTeX file (Approximately 100kb).
3. Postscript file: a zipped English Postscript file (Approximately 384kb).
4. physics.pdf: an English PDF file (776kb).
5. physics-hyperref.pdf: an English PDF file with bookmarks (844kb).
6. fysica.pdf: a Dutch PDF file (783kb).
7. fysica-hyperref.pdf: a Dutch PDF file with bookmarks (855kb).
8. fysica.zip: a zipped Dutch LaTeX file (Approximately 100kb).

## Equations in physics - Contents

```   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
10.1.2 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
11.7 The Boltzmann transport equation                             59
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
14.4 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

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
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