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.
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