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Equations in Physics
By ir
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C
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Wevers
Contents
Contents
I
Physical Constants
1
1 Mechanics
1
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1
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1 Definitions
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1
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1
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1
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1
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1 Force, (angular)momentum and energy
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3
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1
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3 Gravitation
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3
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1
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5 The virial theorem
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4 Point dynamics in a moving coordinate system
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4
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1
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2 Tensor notation
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5 Dynamics of masspoint collections
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5
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1
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2 Collisions
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6 Dynamics of rigid bodies
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6
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1
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2 Principal axes
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6
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1
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7
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1
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2 Hamilton mechanics
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7
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4 Phase space, Liouville’s equation
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7
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2
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2 Electricity & Magnetism
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2
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2
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2
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5 Electromagnetic waves
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5
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5
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6 Multipoles
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7 Electric currents
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8 Depolarizing field
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9 Mixtures of materials
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9
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in mater
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J
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A
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1 Special relativity
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1
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3
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2 Red and blue shift
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1
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2 General relativity
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2
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3
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2 The line element
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2
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3
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4 The trajectory of a photon
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2
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3
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6 Cosmology
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13
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4 Oscillations
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6 Pendulums
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1 The wave equation
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2 Solutions of the wave equation
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2
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5
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2 Spherical waves
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2
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5
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4 The general solution in one dimension
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5
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5 Waveguides and resonating cavities
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6 Non-linear wave equations
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1 The bending of light
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2 Paraxial geometrical optics
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2
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6
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2 Mirrors
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2
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6
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4 Magnification
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3 Matrix methods
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4 Aberrations
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5 Reflection and transmission
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6 Polarization
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7 Prisms and dispersion
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8 Diffraction
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9 Special optical effects
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10 The Fabry-Perot interferometer
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24
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7 Statistical physics
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7
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3 Pressure on a wall
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4 The equation of state
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5 Collisions between molecules
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J
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A
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6
Interaction between molecules
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1 Mathematical introduction
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2 Definitions
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3 Thermal heat capacity
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4 The laws of thermodynamics
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5 State functions and Maxwell relations
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1 Mathematical introduction
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2 Conservation laws
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3 Bernoulli’s equations
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4 Caracterising of flows with dimensionless
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1 Flow boundary layers
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32
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10 Quantum physics
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10
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1 Black body radiation
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1
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10
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3 Electron diffraction
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2 Wave functions
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3 Operators in quantum physics
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4 The uncertaincy principle
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5 The Schr¨odinger equation
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6 Parity
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7 The tunnel effect
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8 The harmonic oscillator
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9 Angular momentum
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10 Spin
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11 The Dirac formalism
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12 Atom physics
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12
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10
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2 Eigenvalue equations
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12
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10
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4 Selection rules
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13 Interaction with electromagnetic fields
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14 Perturbation theory
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J
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A
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15 N-particle systems
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1 General
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1 Introduction
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2 Transport
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3 Elastic collisions
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3
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11
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2 The Coulomb interaction
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3
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11
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4 The center of mass system
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3
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5 Inelastic collisions
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5
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11
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2 Cross sections
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6 Radiation
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7 The Boltzmann transport equation
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8 Collision-radiative models
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9 Waves in plasma’s
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12 Solid state physics
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1 lattice with one kind of atoms
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3
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12
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3 Phonons
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3
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12
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12
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1 Dielectrics
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4
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12
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3 Ferromagnetism
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5 Free electron Fermi gas
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5
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12
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2 Electric conductance
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4 Thermal heat conductivity
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6 Energy bands
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7 Semiconductors
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8 Superconductivity
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12
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2 The Josephson effect
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5 The London equation
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J
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A
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1 Introduction
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1
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13
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2 The Cayley table
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1
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13
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4 Isomorfism and homomorfism; representations
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1
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13
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13
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1 Schur’s lemma
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2
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13
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3 Character
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3 The relation with quantummechanics
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3
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13
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2 Breaking of degeneracy with a perturbation
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3
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13
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4 The direct product of representations
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3
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13
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6 Symmetric transformations of operators, irreducible tensor operators
13
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7 The Wigner-Eckart theorem
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4 Continuous groups
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4
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13
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2 The 3-dimensional rotation group
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4
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13
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1 Vectormodel for the addition of angular momentum
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14 Nuclear physics
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14
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1 Kinetic model
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4
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14
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3 Conservation of energy and momentum in nuclear reactions
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15 Quantum field theory & Particle physics
15
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15
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15
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6 Field functions for spin- 21 particles
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7 Quantization of spin- 21 fields
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8 Quantization of the electromagnetic field
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9 Interacting fields and the S-matrix
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10 Divergences and renormalization
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11 Classification of elementary particles
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12 P and CP-violation
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13 The standard model
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13
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15
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2 Spontaneous symmetry breaking
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VI
Equations in Physics by ir
...
C
...
Wevers
15
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3 Quantumchromodynamics
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14 Pathintegrals
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1 Determination of distances
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2 Brightnes and magnitudes
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3 Radiation and stellar atmospheres
16
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5 Energy production in stars
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94
95
96
96
96
97
97
98
Physical Constants
Name
Number π
Number e
Euler’s constant
Elementary charge
Gravitational constant
Fine-structure constant
Speed of light in vacuum
Permittivity of the vacuum
Permeability of the vacuum
(4πε0 )−1
Symbol
Value
Unit
π
3,14159265
e
2,718281828459
n
P
γ = lim
1/k − ln(n) = 0, 5772156649
n→∞
k=1
e
G, κ
α = e2 /2hcε0
c
ε0
µ0
Planck’s constant
Dirac’s constant
Bohr magneton
Bohr radius
Rydberg’s constant
Electron Compton wavelength
Proton Compton wavelength
Reduced mass of the H-atom
h
¯h = h/2π
µB = e¯h/2me
a0
Ry
λCe = h/me c
λCp = h/mp c
µH
Stefan-Boltzmann’s constant
Wien’s constant
Molar gasconstant
Avogadro’s constant
Boltzmann’s constant
σ
kW
R
NA
k = R/NA
Electron mass
Proton mass
Neutron mass
Elementary mass unit
Nuclear magneton
me
mp
mn
mu =
µN
Diameter of the Sun
Mass of the Sun
Rotational period of the Sun
Radius of Earth
Mass of Earth
Rotational period of Earth
Astronomical unit
Light year
Parsec
Hubble constant
D⊙
M⊙
T⊙
RA
MA
TA
AU
lj
pc
H
1, 60217733 · 10−19
6, 67259 · 10−11
≈ 1/137
2, 99792458 · 108
8, 854187 · 10−12
4π · 10−7
8, 9876 · 109
Js
Js
Am2
˚
A
eV
m
m
kg
9, 1093897 · 10−31
1, 6726231 · 10−27
1, 674954 · 10−27
1, 6605656 · 10−27
5, 0508 · 10−27
kg
kg
kg
kg
J/T
1392 · 106
1, 989 · 1030
25,38
6, 378 · 106
5, 976 · 1024
23,96
1, 4959787066 · 1011
9, 4605 · 1015
3, 0857 · 1016
≈ (75 ± 25)
1
m/s (def)
F/m
H/m
Nm2 C−2
6, 6260755 · 10−34
1, 0545727 · 10−34
9, 2741 · 10−24
0, 52918
13,595
2, 2463 · 10−12
1, 3214 · 10−15
9, 1045755 · 10−31
5, 67032 · 10−8
2, 8978 · 10−3
8,31441
6, 0221367 · 1023
1, 380658 · 10−23
1
12
12 m( 6 C)
C
m3 kg−1 s−2
Wm2 K−4
mK
J/mol
mol−1
J/K
m
kg
days
m
kg
hours
m
m
m
km·s−1 ·Mpc−1
Chapter 1
Mechanics
1
...
1
...
The following holds:
Z
Z
Z
s(t) = s0 + |~v (t)|dt ; ~r(t) = ~r0 + ~v (t)dt ; ~v (t) = ~v0 + ~a(t)dt
When the acceleration is constant this gives: v(t) = v0 + at and s(t) = s0 + v0 t + 21 at2
...
1
...
So, for the unit coordinate vectors holds:
˙eθ , ~e˙θ = −θ~
˙ er
~e˙r = θ~
˙eθ , ~a = (¨
The velocity and the acceleration are derived from: ~r = r~er , ~v = r~
˙ er + rθ~
r − rθ˙2 )~er + (2r˙ θ˙ +
¨ eθ
...
2
Relative motion
For the motion of a point D w
...
t
...
with QD
ω2
¨ ′ means that the quantity is defined in a moving system of coordinates
...
moving system holds:
~v = ~vQ + ~v ′ + ω
~ × ~r ′ and ~a = ~aQ + ~a ′ + α
~ × ~r ′ + 2~ω × ~v − ~ω × (~ω × ~r ′ )
with |~ω × (~
ω × ~r ′ )| = ω 2~rn ′
1
...
3
...
2
3
Chapter 1: Mechanics
˙ = F~ · ~v
...
Z
~ is given by: S
~ = ∆~
The kick S
p = F~ dt
The work A, delivered by a force, is A =
Z
2
1
F~ · d~s =
Z
2
F cos(α)ds
1
~ ~τ = L
~˙ = ~r × F~ ; and
The torque ~τ is related to the angular momentum L:
2
~
~
L = ~r × p~ = m~v × ~r, |L| = mr ω
...
So, the conditions for a mechanical equilibrium are:
τ =−
The force of friction is usually proportional with the force perpendicular to the surface, except when
the motion starts, when a threshold has to be overcome: Ffric = f · Fnorm · ~et
...
3
...
From this follows
A conservative force can be written as the gradient of a potential: F~cons = −∇U
~
~
that rotF = 0
...
1
...
3
Gravitation
The Newtonian law of gravitation is (in GRT one also uses κ instead of G):
m1 m2
F~g = −G 2 ~er
r
The gravitationpotential is then given by V = −Gm/r
...
1
...
4
Orbital equations
From the equations of Lagrange for φ, conservation of angular momentum can be derived:
∂V
d
∂L
=
= 0 ⇒ (mr2 φ) = 0 ⇒ Lz = mr2 φ = constant
∂φ
∂φ
dt
For the radius as a function of time can be found that:
2
dr
L2
2(W − V )
− 2 2
=
dt
m
m r
The angular equation is then:
φ − φ0 =
Zr "
0
mr2
L
r
2(W − V )
L2
− 2 2
m
m r
#−1
dr
r −2 field
=
arccos 1 +
1
r
1
r0
−
1
r0
+ km/L2z
!
if F = F (r): L =constant, if F is conservative: W =constant, if F~ ⊥ ~v then ∆T = 0 and U = 0
...
J
...
A
...
The equation of the orbit
is:
ℓ
r(θ) =
, or: x2 + y 2 = (ℓ − εx)2
1 + ε cos(θ − θ0 )
with
ℓ=
L2
;
Gµ2 Mtot
ε2 = 1 +
2W L2
2
G2 µ3 Mtot
=1−
ℓ
;
a
a=
k
ℓ
=
2
1−ε
2W
a is half the length of the
√ long axis of the elliptical orbit in case the orbit is closed
...
ε is the excentricity of the orbit
...
Now, 5 kinds of orbits are possible:
1
...
2
...
3
...
4
...
5
...
Other combinations are not possible: the total energy in a repulsive force field is always positive so
ε > 1
...
3
...
1
...
4
...
The different apparent forces are given
by:
1
...
Rotation: F~α = −m~
α × ~r
′
3
...
Centrifugal force: F~cf = mω 2~rn ′ = −F~cp ; F~cp = −
~er
r
1
...
2
Tensor notation
Transformation of the Newtonian equations of motion to xα = xα (x) gives:
xβ
∂xα d¯
dxα
=
;
dt
∂x
¯β dt
so
d dxα
d
d2 xα
=
=
2
dt dt
dt
dt
∂xα d¯
xβ
∂x
¯β dt
∂xα d2 x
¯β
d¯
xβ d
=
+
∂x
¯β dt2
dt dt
∂xα
∂x
¯β
The chain rule gives:
xγ
xγ
d ∂xα
∂ ∂xα d¯
∂ 2 xα d¯
=
=
dt ∂ x
¯β
∂x
¯γ ∂ x
¯β dt
∂x
¯β ∂ x
¯γ dt
So:
∂xα d2 x
¯β
∂ 2 xα d¯
xγ
d2 xα
=
+
dt2
∂x
¯β dt2
∂x
¯β ∂ x
¯γ dt
So the Newtonian equation of motion
m
will be transformed into:
m
d2 xα
= Fα
dt2
dxβ dxγ
d2 xα
+ Γα
βγ
2
dt
dt dt
= Fα
The apparent forces are brought from he origin to the effect side in the way Γα
βγ
1
...
5
...
dt dt
Dynamics of masspoint collections
The center of mass
~ is given by ~v − R
...
r
...
the center of mass R
given by:
P
mi~ri
~rm = P
mi
In a 2-particle system, the coordinates of the center of mass are given by:
~ = m1~r1 + m2~r2
R
m1 + m2
With ~r = ~r1 − ~r2 , the kinetic energy becomes: T = 21 Mtot R˙ 2 + 12 µr˙ 2 , with the reduced mass µ is given
1
1
1
by:
+
=
µ
m1
m2
The motion within and outside the center of mass can be separated:
~˙ outside = ~τoutside ;
L
p = m~vm ;
~
~˙ inside = ~τinside
L
F~ext = m~am ;
F~12 = µ~u
6
Equations in Physics by ir
...
C
...
Wevers
1
...
2
Collisions
With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds:
p~ = m~vm is constant, and T = 21 m~vm2 is constant
...
Further holds ∆L
w
...
t
...
1
...
6
...
r
...
D = Iw
...
t
...
Object
I
Object
I
Cavern cylinder
I = mR2
Massive cylinder
I = 21 mR2
Disc, axis in plane disc through m
I = 41 mR2
Halter
I = 21 µR2
Cavern sphere
I = 32 mR2
Massive sphere
I = 52 mR2
Bar, axis ⊥ through c
...
m
...
c
...
m
...
m
I = ma2
1
...
2
+ b2 )
Principal axes
Each rigid body has (at least) 3 principal axes which stand ⊥ at each other
...
6
...
Ik
Time dependence
For torque of force ~τ holds:
~τ ′ = I θ¨ ;
~
The torque T~ is defined by: T~ = F~ × d
...
7
1
...
1
Variational Calculus, Hamilton and Lagrange mechanics
Variational Calculus
Starting with:
δ
Zb
a
L(q, q,
˙ t)dt = 0 met δ(a) = δ(b) = 0 and δ
du
dx
=
d
(δu)
dx
the equations of Lagrange can be derived:
d ∂L
∂L
=
dt ∂ q˙i
∂qi
When there are additional conditions applying on the variational problem δJ(u) = 0 of the type
K(u) =constant, the new problem becomes: δJ(u) − λδK(u) = 0
...
7
...
The Hamiltonian is given by: H = q˙i pi − L
...
If the used coordinates are canonical are the Hamilton equations the equations of motion for the
system:
dpi
∂H
∂H
dqi
;
=
=−
dt
∂pi
dt
∂qi
Coordinates are canonical if the following holds: {qi , qj } = 0, {pi , pj } = 0, {qi , pj } = δij where {, }
is the Poisson bracket:
X ∂A ∂B
∂A ∂B
−
{A, B} =
∂qi ∂pi
∂pi ∂qi
i
1
...
3
Motion around an equilibrium, linearization
For natural systems around equilibrium holds:
2
∂V
∂ V
= 0 ; V (q) = V (0) + Vik qi qk with Vik =
∂qi 0
∂qi ∂qk 0
With T = 12 (Mik q˙i q˙k ) one receives the set of equations M q¨ + V q = 0
...
This leads to the eigenfrequentions
aT V a k
of the problem: ωk2 = Tk
...
The general solution
ak M ak
is a superposition if eigenvibrations
...
7
...
J
...
A
...
7
...
If pi q˙i − H = Pi Qi − K(Pi , Qi , t) −
pi =
dF1 (qi , Qi , t)
, the coordinates follow from:
dt
∂F1
;
∂qi
Pi =
∂F1
;
∂Qi
K =H+
dF1
dt
dF2 (qi , Pi , t)
, the coordinates follow from:
2
...
If −p˙ i qi − H = Pi Q˙ i − K +
∂F2
;
∂qi
Qi =
∂F2
;
∂Pi
K =H+
∂F2
∂t
dF3 (pi , Qi , t)
, the coordinates follow from:
dt
qi = −
4
...
The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by:
H=
2
1
~ + qV
p~ − q A
2m
This Hamiltonian can be derived from the Hamiltonian of a free particle H = p2 /2m with the
~ and H → H − qV
...
A gaugetransformation
on the potentials Aα corresponds with a canonical transformation, which make the Hamilton equations
the equations of motion for the system
...
1
The Maxwell equations
The classical electromagnetic field can be described with the Maxwell equations, and can be written
both as differential and integral equations:
ZZ
~ · ~n)d2 A = Qfree,included
~ = ρfree
(D
∇·D
ZZ
~ · ~n)d2 A = 0
~ =0
(B
∇·B
I
~
~ = − ∂B
~ · d~s = − dΦ
∇×E
E
dt
∂t
I
~
dΨ
~ = J~free + ∂ D
~ · d~s = Ifree,included +
∇×H
H
dt
∂t
ZZ
ZZ
~ · ~n)d2 A, Φ =
~ · ~n)d2 A
...
2
Force and potential
The force and the electric field between 2 point charges are given by:
F~12 =
Q1 Q2
~er ;
4πε0 εr r2
~
~ = F
E
Q
The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field
...
~
The origin of this force is a relativistic transformation of the Coulomb force: F~L = Q(~v × B)
The magnetic field which results from an electric current is given by the law of Biot-Savart:
~ =
dB
µ0 I ~
dl × ~er
4πr2
If the current is time-dependent one has to take retardation into account: the substitution I(t) →
I(t − r/c) has to be applied
...
J
...
A
...
The fields can be derived from the
potentials as follows:
~
~ = ∇×A
~
~ = −∇V − ∂ A , B
E
∂t
~ = ~v × E
...
3
Gauge transformations
The potentials of the electromagnetic fields transform as follows when a gauge transformation is
applied:
~′ = A
~ − ∇f
A
∂f
V′ =V +
∂t
~
~
so the fields E and B do not change
...
Further, the freedom remains to apply a limiting condition
...
This separates the differential equations for A
~ and V :
1
...
2V = − , 2A
ε0
~ = 0
...
2
...
4
Energy of the electromagnetic field
The energy density of the electromagnetic field is:
Z
Z
dW
= w = HdB + EdD
dVol
The energy density can be expressed in the potentials and currents as follows:
Z
Z
~ 3 x , wel = 1 ρV d3 x
wmag = 12 J~ · Ad
2
2
...
5
...
The irradiance
The radiated energy can be derived from the Poynting vector S:
~ t
...
2
...
2
Electromagnetic waves in mater
The wave equations in matter, with cmat = (εµ)−1/2 are:
∂2
∂2
µ ∂ ~
µ ∂ ~
2
2
∇ − εµ 2 −
E=0,
∇ − εµ 2 −
B=0
∂t
ρ ∂t
∂t
ρ ∂t
~ = E exp(i(~k · ~r − ωt)) and B
~ = B exp(i(~k ·
give, after substitution of monochromatic plane waves: E
~r − ωt)) the dispersion relation:
iµω
k 2 = εµω 2 +
ρ
The first term arises from the displacement current, the second from the conductance current
...
If the material is a good
This results in a damped wave: E
r
µω
...
6
Multipoles
∞
Because
1
1X
=
′
|~r − ~r |
r 0
′ l
r
Q X kn
Pl (cos θ) can the potential be written as: V =
r
4πε n rn
For the lowest-order terms this results in:
R
• Monopole: l = 0, k0 = ρdV
R
• Dipole: l = 1, k1 = r cos(θ)ρdV
P
• Quadrupole: l = 2, k2 = 12 (3zi2 − ri2 )
i
~ ext ,
1
...
3~
p · ~r
~ ≈ Q
~ out
Electric field: E
−
p
~
...
The magnetic dipole: dipolemoment: if r ≫ A: ~µ = I~ × (A~e⊥ ), F~ = (~µ · ∇)B
2
mv⊥
~ out
|µ| =
, W = −~
µ×B
2B
~ = −µ 3µ · ~r − ~µ
...
7
Electric currents
The continuity equation for charge is:
∂ρ
+ ∇ · J~ = 0
...
J
...
A
...
dΦ
...
If a conductor encloses a flux Φ holds: Φ = LI
...
The energy contained within a coil is given by W = 12 LI 2 and
L = µN 2 A/l
...
For a capacitor holds: C = ε0 εr A/d where d is the distance
between the plates and A the surface of one plate
...
The accumulated energy is given by W = 12 CV 2
...
The current through a capacity is given by I = −C
dt
For most PTC resistors holds approximately: R = R0 (1 + αT ), where R0 = ρl/A
...
If a current flows through two different, connecting conductors x and y, the contact area will heat up
or cool down, depending on the direction of the current: the Peltier effect
...
This effect can be amplified with semiconductors
...
For a Cu-Konstantane connection
holds: γ ≈ 0, 2 − 0
...
In
Kirchhoff ’s equations apply: for a knot holds:
P an electrical net with only stationary
P currents,
P
In = 0, along a closed path holds:
Vn = In Rn = 0
...
8
Depolarizing field
If a dielectric material is placed in an electric or magnetic field, the field strength within and outside
the material will change because the material will be polarized or magnetized
...
~
~ dep = E
~ mat − E
~0 = − N P
E
ε0
~
~
~
~
Hdep = Hmat − H0 = −N M
N is a constant depending only on the shape of the object placed in the field, with 0 ≤ N ≤ 1
...
2
...
For a sphere holds:
by: hDi = hεEi = ε hEi where ε = ε1 1 −
Φ(ε∗ /ε2 )
2
1
Φ = 3 + 3 x
...
1
3
...
1
Special relativity
The Lorentz transformation
The Lorentz transformation (~x′ , t′ ) = (~x′ (~x, t), t′ (~x, t)) leaves the wave equation invariant if c is
invariant:
∂2
∂2
1 ∂2
∂2
∂2
∂2
1 ∂2
∂2
+ 2+ 2− 2 2 =
+ ′2 + ′2 − 2 ′2
2
′2
∂x
∂y
∂z
c ∂t
∂x
∂y
∂z
c ∂t
This transformation can also be found when ds2 = ds′2 is demanded
...
The proper time τ is defined as:
dτ 2 = ds2 /c2 , so ∆τ = ∆t/γ
...
p = mv = W v/c2 , and pc = W β where β = v/c
...
J
...
A
...
The difference of two 4-vectors
dxα
...
The 4-vector for the velocity is given by U α =
dτ
i
i
α
i
the “common” velocity u := dx /dt is: U = (γu , icγ)
...
The 4-vector for
energy and momentum is given by: pα = m0 U α = (γpi , iW/c)
...
3
...
2
Red and blue shift
There are three causes of red and blue shifts:
f′
v cos(ϕ)
1
...
=γ 1−
f
c
This can give both red- and blueshift, also ⊥ the direction of motion
...
Gravitational redshift:
κM
∆f
= 2
...
Redshift because the universe expands, resulting in e
...
the cosmic background radiation:
R0
...
1
...
The electromagnetic field tensor is given
by:
∂Aβ
∂Aα
Fαβ =
−
∂xα
∂xβ
~ iV /c) and Jµ := (J~, icρ)
...
2
3
...
1
General relativity
Riemannian geometry, the Einstein tensor
The basic principles of general relativity are:
1
...
For particles with zero rest mass
R (photons), the use of a
free parameter is required because for them holds ds = 0
...
The principle of equivalence: inertial mass ≡ gravitational mass ⇒ gravitation is equivalent
with a curved space-time were particles move along geodesics
...
By a proper choice of the coordinate system it is possible to make the metric locally flat in each
point xi : gαβ (xi ) = ηαβ :=diag(−1, 1, 1, 1)
...
Here,
Γijk =
∂ 2 x¯l ∂xi
∂xj ∂xk ∂ x
¯l
µ
σ
are the Christoffel symbols
...
α
α
α
σ
α
σ
The following holds: Rβµν
= ∂µ Γα
βν − ∂ν Γβµ + Γσµ Γβν − Γσν Γβµ
...
The Bianchi identities are: ∇λ Rαβµν + ∇ν Rαβλµ + ∇µ Rαβνλ = 0
...
With the variational principle δ (L(gµν ) − Rc /16πκ) |g|d4 x = 0 for variations
gµν → gµν + δgµν the Einstein field equations can be derived:
Gαβ =
8πκ
Tαβ
c2
, which can also be written as Rαβ =
8πκ
(Tαβ − 21 gαβ Tµµ )
c2
For empty space this is equivalent with Rαβ = 0
...
The Einstein equations are of 10 independent equations, who are second order in gµν
...
In the stationary case, this results in ∇2 h00 = 8πκ̺/c2
...
This constant plays a role in inflatory models of the universe
...
2
...
i
∂x ∂xj
k
In general holds: ds2 = gµν dxµ dxν
...
This metric, ηµν :=diag(−1, 1, 1, 1), is called the Minkowski metric
...
This metric is singular for r = 2m = 2κM/c2
...
The Newtonian limit of this metric is given by:
ds2 = −(1 + 2V )c2 dt2 + (1 − 2V )(dx2 + dy 2 + dz 2 )
where V = −κM/r is the Newtonian gravitation potential
...
The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric
near r = 2m
...
J
...
A
...
The line element in these coordinates is given by:
ds2 = −
32m3 −r/2m 2
e
(dv − du2 ) + r2 dΩ2
r
The line r = 2m corresponds to u = v = 0, the limit x0 → ∞ with u = v and x0 → −∞ with u = −v
...
On the line dv = ±du holds dθ = dϕ = ds = 0
...
3
...
3
Planetary orbits and the perihelium shift
R
To find a planetary
problem δ ds = 0 has to be solved
...
Substituting the external Schwarzschild metric gives for a
planetary orbit:
m
du d2 u
du
3mu
+
+
u
=
dϕ dϕ2
dϕ
h2
where u := 1/r and h = r2 ϕ˙ =constant
...
This
h2
κM
1+ 2
...
In zeroth order, this results in an elliptical orbit: u0 (ϕ) = A + B cos(ϕ) with
A = m/h2 and B an arbitrary constant
...
The perihelion of a planet is the point for which r is minimal, or u
maximal
...
For the perihelion shift then follows:
∆ϕ = 2πε = 6πm2 /h2 per orbit
...
2
...
Substituting
the external Schwarzschild metric results in the following orbital equation:
du d2 u
+ u − 3mu = 0
dϕ dϕ2
3
...
5
Gravitational waves
Starting with the approximation gµν = ηµν + hµν for weak gravitational fields, and the definition
′
′
ν
h′µν = hµν − 21 ηµν hα
α , follows that 2hµν = 0 if the gauge condition ∂hµν /∂x = 0 is satisfied
...
2
...
Cosmology
If for the universe as a whole is assumed:
1
...
The 3-dimensional spaces are isotrope for a certain value of x0 ,
3
...
then the Robertson-Walker metric can be derived for the line element:
ds2 = −c2 dt2 +
R2 (t)
2
2
2
2 (dr + r dΩ )
kr
r02 1 − 2
4r0
For the scalefactor R(t) the following equations can be derived:
¨
8πκp
R˙ 2 + kc2
2R
=− 2
+
2
R
R
c
and
R˙ 2 + kc2
8πκ̺
=
2
R
3
where p is the pressure and ̺ the density of the universe
...
This is a measure of the velocity of which galaxies far away
are moving away of each other, and has the value ≈ (75 ± 25) km·s−1 ·Mpc−1
...
Parabolical universe: k = 0, W = 0, q = 21
...
The hereto related density ̺c = 3H 2 /8πκ is the critical density
...
Hyperbolical universe: k = −1, W < 0, q < 21
...
3
...
The expansion velocity of the universe becomes
negative after some time: the universe starts falling together
...
1
Harmonic oscillations
ˆ i(ωt±ϕ) ≡ Ψ
ˆ cos(ωt ± ϕ),
The general shape of a harmonic oscillation is: Ψ(t) = Ψe
ˆ is the amplitude
...
2
Z
x(t)dt =
X
i
ˆ 2i + 2
Ψ
XX
j>i
i
ˆ j cos(αi − αj )
ˆ iΨ
Ψ
dn x(t)
x(t)
= (iω)n x(t)
...
With complex amplitudes, this becomes −mω 2 x = F − Cx − ikωx
...
The quantity Z = F/x˙ is called the impedance of the system
...
system is given by Q =
k
where δ =
The frequency with minimal
|Z| is called velocity resonance frequency
...
In the
√
resonance curve is |Z|/ Cm plotted
against ω/ω0
...
In these points holds: R = X and δ = ±Q−1 , and the width is
2∆ωB = ω0 /Q
...
The amplitude
resonance frequency ωA is the
q
1 2
frequency where iωZ is minimal
...
The damping frequency
r ωD is a measure for the time in which an oscillating system comes to rest
...
A weak damped oscillation (k 2 < 4mC) dies after TD = 2π/ωD
...
A strong damped oscillation (k 2 > 4mC)
drops like (if k 2 ≫ 4mC) x(t) ≈ x0 exp(−t/τ )
...
3
Electric oscillations
The impedance is given by: Z = R + iX
...
The impedance of
a resistor is R, of a capacitor 1/iωC and of a self inductor iωL
...
The total impedance in case several elements are positioned is given by:
1
...
parallel connection: V = IZ,
X 1
X 1
X
R
1
1
R
Ci , Q =
=
,
=
, Ctot =
, Z=
Ztot
Z
L
L
Z
1
+
iQδ
i
tot
i
0
i
i
i
Here, Z0 =
r
1
L
and ω0 = √
...
4
...
g
...
For them holds: Z0 =
r
dx dx
The transmission velocity is given by v =
...
5
r
dL dx
...
For the coefficients of mutual
induction Mij holds:
M12 = M21 := M = k
p
N 2 Φ2
N 1 Φ1
=
∼ N1 N2
L1 L2 =
I2
I1
where 0 ≤ k ≤ 1 is the coupling factor
...
At full load holds:
r
V1
L1
I2
iωM
N1
=
=−
≈−
=−
V2
I1
iωL2 + Rload
L2
N2
4
...
p
• Physical pendulum: T = 2π I/τ with τ the moment of force and I the moment of inertia
...
p
I/κ with κ =
• Mathematic pendulum: T = 2π
pendulum
...
1
The wave equation
The general shape of the wave equation is: 2u = 0, or:
∇2 u −
∂2u ∂2u ∂2u
1 ∂2u
1 ∂2u
=
+ 2 + 2 − 2 2 =0
2
2
2
v ∂t
∂x
∂y
∂z
v ∂t
where u is the disturbance and v the propagation velocity
...
Per definition
holds: kλ = 2π and ω = 2πf
...
Longitudinal waves: for these holds ~k k ~v k ~u
...
Transversal waves: for these holds ~k k ~v ⊥ ~u
...
The group velocity is given by:
k dn
dvph
dω
= vph + f
= vph 1 −
vg =
dk
dk
n dk
where n is the refractive index of the medium
...
g
...
If vph does not depend on ω
holds: vph = vg
...
For some media, the propagation velocity follows from:
p
• Pressure waves in a liquid or gas: v = κ/̺, where κ is the modulus of compression
...
p
• Pressure waves in a solid bar: v = E/̺
p
• waves in a string: v = Fspan l/m
s
2πh
gλ 2πγ
• Surface waves on a liquid: v =
+
tanh
2π
̺λ
λ
√
where h is the depth of the liquid and γ the surface tension
...
5
...
2
...
A fixed end gives a phase
change of π/2 to the reflected wave, with boundary condition u(l) = 0
...
If an observer is moving w
...
t
...
the Doppler effect
...
2
...
2
...
For sufficient
large values of r these are approximated with:
u
ˆ
u(r, t) = √ cos(k(r ± vt))
r
5
...
4
The general solution in one dimension
Starting point is the equation:
N
X
∂m
∂ 2 u(x, t)
b
u(x, t)
=
m
∂t2
∂xm
m=0
where bm ∈ IR
...
The general solution is given by:
Z∞
a(k)ei(kx−ω1 (k)t) + b(k)ei(kx−ω2 (k)t) dk
u(x, t) =
−∞
Because in general the frequencies ωj are non-linear in k there is dispersion and the solution can not
be written any more as a sum of functions depending only on x ± vt: the wave front transforms
...
3
The stationary phase method
The Fourier integrals of the previous section can usually not be calculated exact
...
Assuming that a(k) is only a slowly varying function of k,
one can state that the parts of the k-axis where the phase of kx−ω(k)t changes rapidly will give no net
contribution to the integral because the exponent oscillates rapidly there
...
dk
Now the following approximation is possible:
v
Z∞
N u
X
u 2π
a(k)ei(kx−ω(k)t) dk ≈
t d2 ω(k ) exp −i 41 π + i(ki x − ω(ki )t)
i
−∞
i=1
dki2
22
Equations in Physics by ir
...
C
...
Wevers
5
...
Starting with the wave equation in one dimension, with ∇2 = ∂ 2 /∂x2 holds: if Q(x, x′ , t)
∂Q(x, x′ , 0)
is the solution with initial values Q(x, x′ , 0) = δ(x − x′ ) and
= 0, and P (x, x′ , t) the
∂t
∂P (x, x′ , 0)
= δ(x − x′ ), then the solution of the wave
solution with initial values P (x, x′ , 0) = 0 and
∂t
∂u(x, 0)
is given by:
equation with arbitrary initial conditions f (x) = u(x, 0) and g(x) =
∂t
u(x, t) =
Z∞
′
′
′
f (x )Q(x, x , t)dx +
−∞
Z∞
g(x′ )P (x, x′ , t)dx′
−∞
P and Q are called the propagators
...
5
′
′
1
2 [δ(x − x − vt) + δ(x − x + vt)]
(
1
if |x − x′ | < vt
2v
0
if |x − x′ | > vt
∂P (x, x′ , t)
∂t
Waveguides and resonating cavities
The boundary conditions at a perfect conductor can be derived from the Maxwell equations
...
One can now deduce that, if Bz and Ez are not ≡ 0:
i
i
∂Bz
∂Bz
∂Ez
∂Ez
Bx =
B
=
k
k
−
εµω
+
εµω
y
εµω 2 − k 2 ∂x
∂y
εµω 2 − k 2 ∂y
∂x
i
i
∂Ez
∂Ez
∂Bz
∂Bz
Ex =
Ey =
k
k
+ εµω
− εµω
εµω 2 − k 2
∂x
∂y
εµω 2 − k 2
∂y
∂x
Now one can distinguish between three cases:
1
...
Boundary condition: Ez |surf = 0
...
Ez ≡ 0: the Transversal Electric modes (TE)
...
∂n surf
For the TE and TM modes this gives an eigenvalue problem for Ez resp
...
For ω < ωℓ , k is imaginary
and the wave is damped
...
In rectangular conductors
the following expression can be found for the cut-off frequency for modes TEm,n of TMm,n :
2
λℓ = p
(m/a)2 + (n/b)2
23
Chapter 5: Waves
3
...
Than holds:
√
k = ±ω εµ and vf = vg , just as if here were no waveguide
...
In a rectangular, 3 dimensional resonating cavity with edges a, b and c are the possible wave numbers
n2 π
n3 π
n1 π
, ky =
, kz =
This results in the possible frequencies f = vk/2π in
given by: kx =
a
b
c
the cavity:
r
n2y
v n2x
n2
f=
+ 2 + 2z
2
2 a
b
c
For a cubic cavity, with a = b = c, the possible number of oscillating modes NL for longitudinal waves
is given by:
4πa3 f 3
NL =
3v 3
Because transversal waves have two possible polarizations holds for them: NT = 2NL
...
6
Non-linear wave equations
The Van der Pol equation is given by:
d2 x
dx
− εω0 (1 − βx2 )
+ ω02 x = 0
2
dt
dt
βx2 can beqignored for very small values of the amplitude
...
The lowest-order instabilities grow with 2 εω0
...
Oscillations on a time scale ∼ ω0−1 can exist
...
If we assume there exist some timescales τn , 0 ≤ τ ≤ N with ∂τn /∂t = εn and if we put
the secular terms 0 we get:
(
)
2
2
dx
d 1 dx
2
1 2 2
= εω0 (1 − βx )
+ 2 ω0 x
dt 2 dt
dt
This is an energy equation
...
If x2 > 1/β, the right-hand
side changes sign and an increase in energy changes into a decrease of energy
...
The Korteweg-De Vries equation is given by:
∂u
∂u
∂3u
+ au
+ b 3 =0
∂t
| {z∂x}
| ∂x
{z }
non−lin
dispersive
This equation is for example a model for ion-acoustic waves in a plasma
...
1
The bending of light
For the refraction at a surface holds: ni sin(θi ) = nt sin(θt ) where n is the refractive index of the
material
...
The refraction of
light in a material is caused by scattering at atoms
...
From
j
this follows that vg = c/(1 + (ne e2 /2ε0 mω 2 ))
...
More general, it is possible to develop n as: n =
...
The path, followed by a lightray in material can be found with Fermat’s principle:
δ
Z2
1
6
...
2
...
For the refraction at a spherical surface with radius R holds:
n2
n1 − n2
n1
−
=
v
b
R
where |v| is the distance of the object and |b| the distance of the image
...
For a double concave lens holds R1 < 0, R2 > 0, for a double convex lens holds
R1 > 0 and R2 < 0
...
For a lens with thickness d and diameter D holds in
good approximation: 1/f = 8(n − 1)d/D2
...
2
...
Spherical aberration can be reduced by not using spherical mirrors
...
The used signs are:
Quantity
R
f
v
b
6
...
3
+
Concave mirror
Concave mirror
Real object
Real image
−
Convex mirror
Convex mirror
Virtual object
Virtual image
Principal planes
The nodal points N of a lens are defined by the figure on the right
...
The plane ⊥ the optical axis through
the principal points is called the principal plane
...
2
...
Further holds: N · Nα = 1
...
The
f-number is defined by f /Dobjective
...
3
Matrix methods
A light ray can be described by a vector (nα, y) with α the angle with the optical axis and y the
distance to the optical axis
...
M is a product of elementary matrices
...
J
...
A
...
Transfer along length l: MR =
1 0
l/n 1
2
...
4
1
0
−D
1
Aberrations
Lenses usually do not give a perfect image
...
Chromatic aberration is caused by the fact that n = n(λ)
...
Using N lenses
makes it possible to obtain the same f for N wavelengths
...
Spherical aberration is caused by second-order effects which are usually ignored; a spherical
surface does not make a perfect lens
...
3
...
Further away of the optical axis they are curved
...
4
...
5
...
6
...
This can be corrected with a
combination of positive and negative lenses
...
5
Reflection and transmission
If an electromagnetic wave hits a transparent medium a part of the wave shall reflect at the same
angle as the incident angle, and a part will be refracted at an angle following from Snell’s law
...
r
...
the surface
...
Then the Fresnel equations
are:
sin(θt − θi )
tan(θi − θt )
, r⊥ =
rk =
tan(θi + θt )
sin(θt + θi )
tk =
2 sin(θt ) cos(θi )
sin(θt + θi ) cos(θt − θi )
,
t⊥ =
2 sin(θt ) cos(θi )
sin(θt + θi )
The following holds: t⊥ − r⊥ = 1 and tk + rk = 1
...
Special is the case r⊥ = 0
...
From Snell’s law then follows: tan(θi ) = n
...
The situation with rk = 0 is not possible
...
6
Polarization
The polarization is defined as: P =
Ip
Imax − Imin
=
Ip + Iu
Imax + Imin
where the intensity of the polarized light is given by Ip and the intensity of the unpolarized light
is given by Iu
...
If polarized light passes through a polarizer Malus law applies: I(θ) = I(0) cos2 (θ) where
θ is the angle of the polarizer
...
The first is independent of the polarization, the second and third are linear
polarizers with the transmission axes horizontal and at +45◦ , while the fourth is a circular polarizer
which is opaque for L-states
...
The state of a polarized light ray can also be described with the Jones vector:
E0x eiϕx
~
E=
E0y eiϕy
~ = (1, 0), for the vertical P -state E
~ = (0, 1), the R-state is given
For the horizontal
P -state holds: E
√
√
1
1
~
~
by E = 2 2(1, −i) and the L-state by E = 2 2(1, i)
...
For some kinds of optical equipment
passage of optical equipment can be described as E
the Jones matrix M is given by:
1 0
Horizontal linear polarizer:
0 0
0 0
Vertical linear polarizer:
0 1
1 1
1
Linear polarizer at +45◦
2
1 1
1 −1
1
◦
Lineair polarizer at −45
2
−1 1
1 0
1
iπ/4
-λ
plate,
fast
axis
vertical
e
4
0 −i
1 0
1
iπ/4
-λ
plate,
fast
axis
horizontal
e
4
0 i
1 i
1
Homogene circulair polarizor right
2
−i 1
1 −i
1
Homogene circular polarizer left
2
i 1
6
...
r
...
the incident direction, where α is the apex angle, θi is the angle between the
incident angle and a line perpendicular to the surface and θi′ is the angle between the ray leaving the
prism and a line perpendicular to the surface
...
For the refractive index of the prism now holds:
n=
sin( 21 (δmin + α))
sin( 21 α)
The dispersion of a prism is defined by:
D=
dδ dn
dδ
=
dλ
dn dλ
28
Equations in Physics by ir
...
C
...
Wevers
where the first factor depends on the shape and the second on the composition of the prism
...
The
refractive index in this area can usually be approximated by Cauchy’s formula
...
8
Diffraction
Fraunhofer diffraction occurs far away of the source(s)
...
N is the number of slits, b is the width of a slit and d is the
distance between the slits
...
The diffraction through a spherical aperture with radius a is described by:
2
J1 (ka sin(θ))
I(θ)
=
I0
ka sin(θ)
The diffraction pattern of a rectangular aperture at distance R with length a in the x-direction and
b in the y-direction is described by:
2
2
sin(α′ )
sin(β ′ )
I(x, y)
=
I0
α′
β′
where α′ = kax/2R and β ′ = kby/2R
...
Close at the source the Fraunhofermodel is unusable because it ignores the angle-dependence of
the reflected waves
...
Diffraction limits the resolution of a system
...
For
a circular slit holds: ∆θmin = 1, 22λ/D where D is the diameter of the slit
...
The minimum difference between two wavelengths that gives a separated diffraction
pattern in a multiple slit geometry is given by ∆λ/λ = nN where N is the number of lines and n the
order of the pattern
...
9
Special optical effects
~ is not parallel with E
~ if the polarizability P~ of a material is
• Birefringe and dichroism
...
There are at least 3 directions, the principal axes, in which they are
parallel
...
In case n2 = n3 6= n1 , which happens e
...
at trigonal, hexagonal and tetragonal crystals there
is one optical axis in the direction of n1
...
The extraordinary wave is linear polarized in the plane through the transmission
29
Chapter 6: Optics
direction and the optical axis
...
Double images occur when the incident ray makes
an angle with optical axis: the extraordinary wave will refract, the ordinary will not
...
Incident light will have a phase shift of ∆ϕ =
2πd(|n0 − ne |)/λ0 if an uniaxial crystal is cut in such a way that the optical axis is parallel with
the front and back plane
...
For a quarter-wave plate holds: ∆ϕ = π/2
...
In that case, the optical axis is parallel to E
...
If
the electrodes have an effective length ℓ and are separated by a distance d, the retardation is
given by: ∆ϕ = 2πKℓV 2 /d2 , where V is the applied voltage
...
These crystals are also piezoelectric: their
~ The retardation
polarization changes when a pressure is applied and vice versa: P~ = pd+ε0 χE
...
• The Faraday effect: the polarization of light passing through material with length d and on
which a magnetic field is applied in the propagation direction is rotated by an angle β = VBd
where V is the Verdet constant
...
The radiation is emitted
within a cone with an apex angle α with sin(α) = c/cmedium = c/nvq
...
10
The Fabry-Perot interferometer
For a Fabry-Perot interferometer holds
in general: T + R + A = 1 where T
is the transmission factor, R the reflection factor and A the absorption factor
...
PP
PP
q
d
Screen
Focussing lens
√
√
The width of the peaks at half height is given bt γ = 4/ F
...
The maximum resolution is then given by ∆fmin = c/2ndF
...
1
Degrees of freedom
A molecule consisting of n atoms has s = 3n degrees of freedom
...
A linear molecule has 2 rotational degrees of freedom and a non-linear molecule 3
...
So, for linear molecules this results in a total of s = 6n − 5
...
The average energy of a molecule in thermodynamic equilibrium is hEtot i = 12 skT
...
The rotational and vibrational energy of a molecule are:
Wrot =
¯2
h
l(l + 1) = Bl(l + 1) , Wvib = (v + 21 )¯
hω 0
2I
The vibrational levels are excited if kT ≈ ¯hω, the rotational levels of a hetronuclear molecule are
excited if kT ≈ 2B
...
7
...
The average velocity is given by
√ 2kT /m
hvi = 2α/ π, and v 2 = 23 α2
...
Even s: s = 2l: c(s) =
1
(l − 1)!
2
...
3
Pressure on a wall
The number of molecules that collides with a wall with surface A within a time τ is given by:
ZZZ
3
d N=
Z∞ Zπ Z2π
0
0
nAvτ cos(θ)P (v, θ, ϕ)dvdθdϕ
0
From this follows for the particle flux on the wall: Φ = 41 n hvi
...
4
The equation of state
If intermolecular forces and the own volume of the molecules can be neglected can for gases from
p = 23 n hEi and hEi = 32 kT be derived:
pV = ns RT =
1
N m v2
3
Here, ns is the number of moles particles and N is the total number of particles within volume V
...
In the Van der Waals equation this
corresponds with the critical temperature, pressure and volume of the gas
...
From dp/dV = 0 and d2 p/dV 2 = 0 follows:
Tcr =
a
8a
, Vcr = 3bns
, pcr =
27bR
27b2
For the critical point holds: pcr Vm,cr /RTcr = 83 , which differs from the value of 1 which follows from
the general gas law
...
A virial development is used for even more accurate contemplations:
p(T, Vm ) = RT
B(T ) C(T )
1
+
+
+
·
·
·
Vm
Vm2
Vm3
The Boyle temperature TB is the temperature for which the 2nd virial coefficient is 0
...
The inversion temperature Ti = 2TB
...
5
Equations in Physics by ir
...
C
...
Wevers
Collisions between molecules
The collision probability of a particle in a gas that is translated over a distance dx is given bt nσdx,
p
v1
where σ is the cross section
...
If m1 = m2 holds:
velocity between the particles
...
This means that the average time between two collisions is given by τ =
ℓ=
...
The average
distance between two molecules is 0, 55n−1/3
...
For the average motion of a particle with radius R can be
derived: x2i = 31 r2 = kT t/3πηR
...
pThe equilibrium
holds
√ for a vessel which has a hole with surface A in it√for which √
√ condition
that ℓ ≫ A/π is: n1 T1 = n2 T2
...
If two plates move along each other at a distance d with velocity wx the viscosity η is given by:
Awx
Fx = η
...
It can
d
be derived that η = 31 ̺ℓ hvi where v is the thermal velocity
...
It can be derived that κ = 31 CmV nℓ hvi /NA
...
A better expression for κ can be obtained with the Eucken correction:
κ = (1 + 9R/4cmV )CV · η with an error <5%
...
6
Interaction between molecules
For dipole interaction between molecules can be derived that U ∼ −1/r6
...
This force can be described by Urep ∼ exp(−γr) or Vrep = +Cs /rs with 12 ≤ s ≤ 20
...
The following holds: D ≈ 0, 89rm
...
A more simple model for intermolecular forces assumes a potential U (r) = ∞ for r < D, U (r) = ULJ
for D ≤ r ≤ 3D and U (r) = 0 for r ≥ 3D
...
D
with F (r) the spatial distribution function in spherical coordinates, which is for a homogeneous
distribution given by: F (r)dr = 4nπr2 dr
...
1
Mathematical introduction
If there exists a relation f (x, y, z) = 0 between 3 variables, one can write: x = x(y, z), y = y(x, z)
and z = z(x, y)
...
A homogeneous function of degree m obeys: εm F (x, y, z) = F (εx, εy, εz)
...
2
Definitions
1
p
1
V
• The isochoric pressure coefficient: βV =
• The isothermal compressibility: κT = −
• The isobaric volume coefficient: γp =
1
V
1
• The adiabatic compressibility: κS = −
V
∂p
∂T
V
∂V
∂p
∂V
∂T p
∂V
∂p
T
S
For an ideal gas follows: γp = 1/T , κT = 1/p and βV = −1/V
...
3
Thermal heat capacity
∂S
∂T X
∂H
• The specific heat at constant pressure: Cp =
∂T p
• The specific heat at constant X is: CX = T
• The specific heat at constant volume: CV =
33
∂U
∂T
V
34
Equations in Physics by ir
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...
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For an ideal gas holds: Cmp − CmV = R
...
So Cp = 12 (s + 2)R
...
For a lower T one needs only to consider the thermalized
degrees of freedom
...
In general holds:
Cp − CV = T
∂p
∂T
V
·
∂V
∂T
p
= −T
∂V
∂T
2
p
∂p
∂V
T
≥0
Because (∂p/∂V )T is always < 0, the following is always true: Cp ≥ CV
...
8
...
The first law is the conservation
of energy
...
In differential form this becomes: d Q = dU + d W ,
where d means that the it is not a differential of a quantity of state
...
So for a reversible process holds: d Q = dU + pdV
...
One can extract a
work Wt from the system or add a work Wt = −Wi to the system
...
So, the entropy difference after
a reversible process is:
Z2
d Qrev
S2 − S1 =
T
1
So, for a reversible cycle holds:
For an irreversible cycle holds:
I
I
d Qrev
= 0
...
T
The third law of thermodynamics is (Nernst):
lim
T →0
∂S
∂X
=0
T
From this can be concluded that the thermal heat capacity → ∞ if T → 0, so absolute zero temperature can not be reached by cooling with a finite number of steps
...
5
State functions and Maxwell relations
The quantities of state and their differentials are:
Internal energy:
Enthalpy:
Free energy:
Gibbs free enthalpy:
U
H = U + pV
F = U − TS
G = H − TS
dU = T dS − pdV
dH = T dS + V dp
dF = −SdT − pdV
dG = −SdT + V dp
35
Chapter 8: Thermodynamics
From this one can derive Maxwell’s relations:
∂p
∂T
∂V
∂p
∂S
∂V
∂S
∂T
=−
,
=
,
=
,
=−
∂V S
∂S V
∂p S
∂S p
∂T V
∂V T
∂T p
∂p T
From the complete differential and the definitions of CV and Cp can be derived that:
∂p
∂V
T dS = CV dT + T
dV and T dS = Cp dT − T
dp
∂T V
∂T p
For an ideal gas also holds:
V
T
p
T
′
+ R ln
+ Sm0 and Sm = Cp ln
− R ln
+ Sm0
Sm = CV ln
T0
V0
T0
p0
Helmholtz’ equations are:
∂U
∂p
∂H
∂V
=T
−p ,
=V −T
∂V T
∂T V
∂p T
∂T p
When a surface is enlarged holds: d Wrev = −γdA, with γ the surface tension
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6
Processes
The efficiency η of a process is given by: η =
Work done
Heat added
The Cold factor ξ of a cooling down process is given by: ξ =
Cold delivered
Work added
Reversible adiabatic processes
For adiabatic processes holds: W = U1 − U2
...
Also holds: T V γ−1 =constant and
T γ p1−γ =constant
...
Isobaric processes
Here holds: H2 − H1 =
The throttle process
R2
1
Cp dT
...
This is also called the Joule-Kelvin effect, and is an adiabatic expansion of a gas through a porous
material or a small opening
...
In general this is acompanied
with a change in temperature
...
Ti = 2TB , with for TB :
[∂(pV )/∂p]T = 0
...
g
...
The Carnotprocess
The system undergoes a reversible cycle with 2 isothemics and 2 adiabatics:
1
...
The system absorbs a heat Q1 from the reservoir
...
Adiabatic expansion with a temperature drop to T2
...
J
...
A
...
Isothermic compression at T2 , removing Q2 from the system
...
Adiabatic compression to T1
...
If the process
is applied in reverse order and the system performs a work −W the cold factor is given by:
ξ=
|Q2 |
T2
|Q2 |
=
=
W
|Q1 | − |Q2 |
T1 − T2
The Stirling process
Stirling’s cycle exists of 2 isothermics and 2 isochorics
...
8
...
The maximum work which can be obtained from this change is,
when all processes are reversible:
1
...
2
...
The minimal work needed to reach a certain state is: Wmin = −Wmax
...
8
Phase transitions
Phase transitions are isothermic and isobaric, so dG = 0
...
The
following holds: rβα = rαβ and rβα = rγα − rγβ
...
In a two phase system holds Clapheyron’s equation:
β
rβα
dp
S α − Sm
=
= m
β
α
α
dT
Vm − Vm
(Vm − Vmβ )T
For an ideal gas one finds for the vapor line at some distance from the critical point:
p = p0 e−rβα/RT
There exist also phase transitions with rβα = 0
...
These second-order transitions appear at organization phenomena
...
g
...
g
...
37
Chapter 8: Thermodynamics
8
...
Because addition of matter usually happens at constant p and T , G is the relevant quantity
...
This is a partial quantity
...
The following holds:
X
xi Vi
Vm =
i
0
=
X
xi dVi
i
where xi = ni /n is the molar fraction of component i
...
The thermodynamic
potentials are not independent in a multiple-phase
system
...
i
i
Each component has as much µ’s as there are phases
...
8
...
i
For the thermodynamic potentials holds: µi = µ0i + RT ln(xi ) < µ0i
...
In spite of this holds
Raoult’s law for the vapor pressure holds for many binary mixtures: pi = xi p0i = yi p
...
A solution of one component in another gives rise to an increase in the boiling point ∆Tk and a
decrease of the freezing point
...
For x2 ≪ 1 holds:
∆Tk =
RT 2
RTk2
x2 , ∆Ts = − s x2
rβα
rγβ
with rβα the evaporation heat and rγβ < 0 the melting heat
...
8
...
In equilibrium
β
γ
holds for each component: µα
i = µi = µi
...
12
Equations in Physics by ir
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Statistical basis for thermodynamics
The number of possibilities P to distribute N particles on n possible energy levels, each with a g-fold
degeneracy is called the thermodynamic probability and is given by:
P = N!
Y g ni
i
i
ni !
The most probable distribution, the one with the maximum value for P , is the equilibrium state
...
The occupation numbers in equilibrium are then given by:
N
Wi
ni = gi exp −
Z
kT
P
The state sum Z is a norming constant, given by: Z = gi exp(−Wi /kT )
...
For a system in thermodynamic equilibrium this
becomes:
N
U
Z
Z
U
≈
+ kN ln
+ k ln
S=
T
N
T
N!
V (2πmkT )3/2
3
5
For an ideal gas, with U = 2 kT then holds: S = 2 kN + kN ln
N h3
8
...
To do this the term
d W = pdV has to be replaced with the correct work term, like d Wrev = −F dl for the stretching of a
wire, d Wrev = −γdA for the expansion of a soap bubble or d Wrev = −BdM for a magnetic system
...
1
Mathematical introduction
An important relation is: if X is a quantity of a volume element which travels from position ~r to
~r + d~r in a time dt, the total differential dX is then given by:
dX =
∂X
∂X
∂X
dX
∂X
∂X
∂X
∂X
∂X
dx +
dy +
dz +
dt ⇒
=
vx +
vy +
vz +
∂x
∂y
∂z
∂t
dt
∂x
∂y
∂z
∂t
dX
∂X
=
+ (~v · ∇)X
...
Some properties of the ∇ operator are:
div(φ~v ) = φdiv~v + gradφ · ~v
div(~u × ~v ) = ~v · (rot~u) − ~u · (rot~v )
div gradφ = ∇2 φ
rot(φ~v ) = φrot~v + (gradφ) × ~v
rot rot~v = grad div~v − ∇2~v
∇2~v ≡ (∇2 v1 , ∇2 v2 , ∇2 v3 )
rot gradφ = ~0
div rot~v = 0
Here, ~v is an arbitrary vector field and φ an arbitrary scalar field
...
2
RR
I
I
(φ · ~et )ds =
(~v · ~et )ds =
ZZ
ZZ
(~n × gradφ)d2 A
(rot~v · ~n)d2 A
ZZ
(rot~v · ~n)d2 A = 0
ZZ
ZZZ
2
(~n × ~v )d A =
(rot~v )d3 A
ZZ
ZZZ
2
(φ~n)d A =
(gradφ)d3 V
d2 A is limited by the Jordan curve
H
ds
...
The force f~0 on each volume element
...
2
...
For these holds: ~t = ~n T, where T is the pressure
tensor
...
J
...
A
...
When viscous aspects can be ignored holds:
divT= −gradp
...
L = D + W
with
1 ∂vi
∂vj
∂vj
1 ∂vi
, Wij :=
+
−
Dij :=
2 ∂xj
∂xi
2 ∂xj
∂xi
When the rotation or vorticity ~
ω = rot~v is introduced holds: Wij = 12 εijk ωk
...
~
rotation velocity: dr
2
For a Newtonian liquid holds: T′ = 2ηD
...
These is related with the
shear stress τ by:
∂vi
τij = η
∂xj
For compressible media can be stated: T′ = (η ′ div~v )I + 2ηD
...
If the viscosity is constant holds: div(2D) =
∇2~v + grad div~v
...
They are:
Integral notation:
∂
1
...
Conservation of momentum:
̺~v d V + ̺~v (~v · ~n)d A =
f0 d V + ~n · T d2 A
∂t
ZZZ
ZZ
∂
3
...
Conservation of mass:
∂̺
+ div · (̺~v ) = 0
∂t
2
...
Conservation of energy: ̺T
∂~v
+ (̺~v · ∇)~v = f~0 + divT = f~0 − gradp + divT′
∂t
de p d̺
ds
=̺ −
= −div~q + T′ : D
dt
dt
̺ dt
Here, e is the internal energy per unit of mass E/m and s is the entropy per unit of mass S/m
...
Further holds:
~q = −κ∇T
p=−
∂e
∂E
=−
,
∂V
∂1/̺
so
CV =
∂e
∂T
V
T =
and Cp =
∂E
∂e
=
∂S
∂s
∂h
∂T
p
41
Chapter 9: Transport phenomena
with h = H/m the enthalpy per unit of mass
...
The force F~ on an object within a flow, when viscous effects are
limited to the boundary layer, can be obtained using the momentum law
...
3
Bernoulli’s equations
Starting with the momentum equation one can find for a non-viscous medium for stationary flows,
with
(~v · grad)~v = 21 grad(v 2 ) + (rot~v ) × ~v
and the potential equation ~g = −grad(gh) that:
Z
dp
1 2
= constant along a streamline
2 v + gh +
̺
For compressible flows holds: 12 v 2 + gh + p/̺ =constant along aRline of flow
...
For
incompressible flows this becomes: 12 v 2 + gh + p/̺ =constant everywhere
...
4
Caracterising of flows with dimensionless numbers
The advantage of dimensionless numbers is that they make model experiments possible: one has
to make the dimensionless numbers which are important for the specific experiment equal for both
model and the real situation
...
Some dimensionless numbers are given by:
v
c
ωL
Strouhal: Sr =
v
a
Fourier:
Fo =
ωL2
ν
Prandtl: Pr =
a
v2
Eckert:
Ec =
c∆T
Mach:
Ma =
vL
ν
v2
Fr =
gL
vL
Pe =
a
Lα
Nu =
κ
Reynolds: Re =
Froude:
P´eclet:
Nusselt:
42
Equations in Physics by ir
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...
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Here, ν = η/̺ is the kinematic viscosity, c is the speed of sound and L is a characteristic length of
the system
...
These numbers can be interpreted as follows:
• Re: (stationary inertial forces)/(viscous forces)
• Sr: (instationary inertial forces)/(stationary inertial forces)
• Fr: (stationary inertial forces)/(gravity)
• Fo: (heat conductance)/(instationary change in enthalpy)
• Pe: (convective heat transport)/(heat conductance)
• Ec: (viscous dissipation)/(convective heat transport)
• Pr and Nu are related to specific materials
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5
~g
∇′2~v
∂~v ′
+ (~v ′ · ∇′ )~v ′ = −grad′ p +
+
′
∂t
Fr
Re
′
Tube flows
For tube flows holds: they are laminar if Re< 2300 with as dimension of length the diameter of the
tube, and turbulent if Re is larger
...
500 < ReD < 2300: Le /2R = 0, 056ReD
2
...
For the total force on a sphere with radius
R in a flow then holds: F = 6πηRv
...
9
...
In the incompressible case
follows from conservation of mass ∇2 φ = 0
...
In general holds:
∂2ψ
∂2ψ
+
= −ωz
2
∂x
∂y 2
In polar coordinates holds:
vr =
∂φ
∂ψ
1 ∂φ
1 ∂ψ
=
, vθ = −
=
r ∂θ
∂r
∂r
r ∂θ
For source flows with power Q in (x, y) = (0, 0) holds: φ =
Q
ln(r) so that vr = Q/2πr, vθ = 0
...
For a vortex holds: φ = Γθ/2π
...
The statement that Fx = 0 is
d’Alembert’s paradox and originates from the neglection of viscous effects
...
So also rotating bodies create a force perpendicular to
their direction of motion: the Magnus effect
...
7
9
...
1
Boundary layers
Flow boundary layers
√
If for the thickness of the boundary layer holds: δ ≪ L holds: δ ≈ L/ Re
...
Blasius’ equation for the
boundary layer is, with vy /v∞ = f (y/δ): 2f ′′′ + f f ′′ = 0 with boundary conditions f (0) = f ′ (0) = 0,
f ′ (∞) = 1
...
The momentum theorem of Von Karman for the boundary layer is:
d
dv
τ0
(ϑv 2 ) + δ ∗ v
=
dx
dx
̺
where the displacement thickness δ ∗ v and the momentum thickness ϑv 2 are given by:
Z∞
ϑv = (v − vx )vx dy ,
2
0
Z∞
∂vx
δ v = (v − vx )dy and τ0 = −η
∂y y=0
∗
0
The boundary layer is released from the surface if
12ηv∞
...
7
...
This is equivalent with
y=0
dp
=
dx
Temperature boundary layers
If the thickness of the temperature boundary layer δT ≪ L holds:
√
1
...
2
...
44
9
...
J
...
A
...
If Φ = 0 the solutions for harmonic oscillations at x = 0 are:
x
T − T∞
x
= exp −
cos ωt −
Tmax − T∞
D
D
p
with D = 2κ/ω̺c
...
The
one-dimensional solution at Φ = 0 is
1
x2
T (x, t) = √
exp −
4at
2 πat
This is mathematical equivalent with the diffusion problem:
∂n
= D∇2 n + P − A
∂t
where P is the production of and A the discharge of particles
...
9
...
For the velocity of the particles holds: v(t) = hvi + v ′ (t) with hv ′ (t)i = 0
...
Boussinesq’s assumption is: τij = −̺ vi′ vj′
...
Near a boundary holds: νt = 0, far
away of a boundary holds: νt ≈ νRe
...
10
Self organization
dω
∂ω
=
+ J(ω, ψ) = ν∇2 ω
dt
∂t
With J(ω, ψ) the Jacobian
...
Further, the kinetic energy/mA and the
enstrofy V are conserved: with ~v = ∇ × (~kψ)
For a (semi) two-dimensional flow holds:
2
E ∼ (∇ψ) ∼
Z∞
0
2
2
E(k, t)dk = constant , V ∼ (∇ ψ) ∼
Z∞
0
k 2 E(k, t)dk = constant
From this follows that in a two-dimensional flow the energy flux goes towards large values of k: larger
structures become larger at the expanse of smaller ones
...
Chapter 10
Quantum physics
10
...
1
...
Wien’s law
for the maximum can also be derived from this: T λmax = kW
...
1
...
1
...
This wavelength is called the Broglie-wavelength
...
2
Wave functions
The wave character of particles is described with a wavefunction ψ
...
Both definitions are each others Fourier transformed:
Z
Z
1
1
Ψ(x, t)e−ikx dx and Ψ(x, t) = √
Φ(k, t)eikx dk
Φ(k, t) = √
h
h
These waves define a particle with group velocity vg = p/m and energy E = h
¯ ω
...
The expectation value hf i of a quantity f of a system is given by:
ZZZ
ZZZ
hf (t)i =
Ψ∗ f Ψd3 V , hfp (t)i =
Φ∗ f Φd3 Vp
This is also written as hf (t)i = hΦ|f |Φi
...
10
...
These operators are hermitian
because their eigenvalues must be real:
Z
Z
ψ1∗ Aψ2 d3 V = ψ2 (Aψ1 )∗ d3 V
45
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Equations in Physics by ir
...
C
...
Wevers
When un is the eigenfunction of the eigenvalue
equation AΨ = aΨ for eigenvalue an , Ψ can be
P
developed to a basis of eigenfunctions: Ψ = cn un
...
If the system is in a state described by Ψ, the chance to find
eigenvalue an when measuring A is given by |cn |2 in the discrete part of the spectrum and |cn |2 da
in the continuous part of the spectrum between a and P
a + da
...
Because (AB)ij = hui |AB|uj i = hui |A |un i hun |B|uj i holds:
|un ihun | = 1
...
For hermitian operators the commutator is
always complex
...
By applying
this to px and x follows (Ehrenfest):
d2
dU (x)
m 2 hxit = −
dt
dx
The first order approximation hF (x)it ≈ F (hxi), with F = −dU/dx represents the classical equation
...
4
The uncertaincy principle
If the uncertaincy ∆A in A is defined as: (∆A)2 = ψ|Aop − hAi |2 ψ = A2 − hAi2 follows:
∆A · ∆B ≥ 12 | hψ|[A, B]|ψi |
From this follows: ∆E · ∆t ≥ 12 h
¯ , and because [x, px ] = i¯h holds: ∆px · ∆x ≥ 21 h
¯ , and ∆Lx · ∆Ly ≥
1
h
¯
L
...
5
The Schr¨
odinger equation
The momentum operator is given by: pop = −i¯h∇
...
The energy
operator is given by: Eop = i¯
h∂/∂t
...
From Hψ = Eψ then follows the Schr¨
odinger
equation:
−
∂ψ
¯2 2
h
∇ ψ + U ψ = Eψ = i¯h
2m
∂t
The linear combination of the solutions of this equation form the general solution
...
6
Parity
The parity operator in one dimension is given by Pψ(x) = ψ(−x)
...
The functions ψ + = 21 (1+P)ψ(x, t) and ψ − = 21 (1−P)ψ(x, t) both satisfy the Schr¨odinger
equation
...
10
...
The energylevels are given by En = n2 h2 /8a2 m
...
If 1, 2 and 3 are the areas before, within and
after the potential well, holds:
ψ1 = Aeikx + Be−ikx ,
′
′
ψ2 = Ceik x + De−ik x ,
ψ3 = A′ eikx
with k ′2 = 2m(W − W0 )/¯
h2 and k 2 = 2mW
...
The amplitude T of the transmitted wave is defined by T = |A′ |2 /|A|2
...
10
...
The Hamiltonian H is then given by:
H=
with
p2
+ 1 mω 2 x2 = 21 h
¯ ω + ωA† A
2m 2
q
ip
ip
+√
and A† = 12 mωx − √
2mω
2mω
†
†
A 6= A is non hermitian
...
A is a raising ladder operator, A† a lowering
ladder operator
...
There is an eigenfunction u0 for which holds: Au0 = 0
...
For the normalized eigenfunctions
follows:
r
† n
mωx2
A
mω
1
4
√
exp −
u0 with u0 =
un = √
π¯h
2¯
h
n!
h
¯
1
with En = ( 2 + n)¯
hω
...
9
q
1
2 mωx
Angular momentum
For the angular momentum operators L holds: [Lz , L2 ] = 0 and [L, H] = 0
...
Not all components of L can be known at the same time with arbitrary accuracy
...
Now holds: L2 = L+ L− + L2z − ¯hLz
...
From [L− , Lz ] = h
¯ L− follows: Lz (L− Ylm ) = (m − 1)¯
h(L− Ylm )
...
Because Lx and Ly are hermitian (this implies L†± = L∓ ) and |L± Ylm |2 > 0 follows: l(l+1)−m2 −m ≥
0 ⇒ −l ≤ m ≤ l
...
Half-odd integral values
give no unique solution ψ and are therefore dismissed
...
10
Equations in Physics by ir
...
C
...
Wevers
Spin
For the spin operators are defined by their commutation relations: [Sx , Sy ] = i¯hSz
...
Because [L, S] = 0 spin and
angular momentum operators do not have a common set of eigenfunctions
...
Then the
probability to find spin up after a measurement is given by |α+ |2 and the chance to find spin down
is given by |α− |2
...
~ by virtue of its spin, given by M
~ =
The electron will have an intrinsic magnetic dipole moment M
~
with gS = 2(1 + α/2π + · · ·) the gyromagnetic ratio
...
~ The Schr¨odinger equation then becomes
magnetic field this gives a potential energy U = −M
because ∂χ/∂xi ≡ 0:
∂χ(t)
eg¯h
~
i¯h
=
~σ · Bχ(t)
∂t
4m
~ = B~ez there are two eigenvalues for this problem: χ± for E =
with ~σ = (~
~σ x , ~
~σ y , ~
~σ z )
...
So the general solution is given by χ = (ae−iωt , beiωt )
...
Thus the spin precesses about the z-axis with
derived: hSx i = 2 ¯
frequency 2ω
...
The potential operator for two particles with spin ± 21 ¯h is given by:
V (r) = V1 (r) +
1 ~ ~
(S1 · S2 )V2 (r) = V1 (r) + 12 V2 (r)[S(S + 1) − 23 ]
h2
¯
This makes it possible for two states to exist: S = 1 (triplet) or S = 0 (Singlet)
...
11
The Dirac formalism
If the operators for p and E are substituted in the relativistic E 2 = m20 c4 + p2 c2 follows the KleinGordon equation:
m 2 c2
1 ∂2
ψ(~x, t) = 0
∇2 − 2 2 − 02
c ∂t
¯h
The operator 2 − m20 c2 /¯
h2 can be separated:
1 ∂2
m20 c2
m20 c2
m20 c2
∂
∂
2
∇ − 2 2−
−
+
γµ
= γλ
c ∂t
∂xλ
∂xµ
h2
¯
¯2
h
¯2
h
where the Dirac matrices γ are given by: γλ γµ + γµ γλ = 2δλµ
...
49
Chapter 10: Quantum physics
10
...
12
...
The Lji are the associated Laguere functions and the
Plm are the associated Legendre polynomials:
|m|
Pl
(x) = (1 − x2 )m/2
d|m| 2
(−1)m n! −x −m dn−m −x n
l
m
(e x )
e x
(x
−
1)
,
L
(x)
=
n
(n − m)!
dxn−m
dx|m|
The parity of these solutions is (−1)l
...
l=0
10
...
2
Eigenvalue equations
The eigenvalue equation for an atom or ion with with one electron are:
Equation
10
...
3
Eigenvalue
4
Range
2
/8ε20 h2 n2
Hop ψ = Eψ
En = µe Z
Lzop Ylm = Lz Ylm
Lz = ml ¯h
L2op Ylm = L2 Ylm
L2 = l(l + 1)¯
h2
Szop χ = Sz χ
Sz = ms ¯h
2
Sop
χ = S2χ
S 2 = s(s + 1)¯
h2
n≥1
−l ≤ ml ≤ l
l
s=
1
2
Spin-orbit interaction
~ +M
~
...
J has quantum
electron
...
, 0,
...
If
~ · L
...
With gS = 2 follows for the average magnetic
~ gem = −(e/2me)g¯
~ where g is the Land´e-factor:
moment: M
hJ,
g =1+
~ · J~
j(j + 1) + s(s + 1) − l(l + 1)
S
=1+
J2
2j(j + 1)
For atoms with more than one electron the following limiting situations occur:
50
Equations in Physics by ir
...
C
...
Wevers
1
...
J ∈ {|L − S|,
...
, J − 1, J}
...
2S + 1 is the multiplicity of a multiplet
...
j − j coupling: for larger atoms is the electrostatic interaction smaller then the Li · si interaction
of an electron
...
jn , J, mJ where only the ji of the not completely
filled subshells are to be taken into account
...
For a transition between two singlet states the line splits in 3 parts, for
∆mJ = −1, 0 + 1
...
At higher S the line splits up in more
parts: the anomalous Zeeman effect
...
10
...
4
Selection rules
~ · ~r |l1 m1 i|
...
For an atom where L − S coupling is dominant further holds: ∆S = 0 (but not strict), ∆L = 0, ±1,
∆J = 0, ±1 except for J = 0 → J = 0 transitions, ∆mJ = 0, ±1, but ∆mJ = 0 is forbidden if
∆J = 0
...
For the total atom holds: ∆J = 0, ±1
but no J = 0 → J = 0 transitions and ∆mJ = 0, ±1, but ∆mJ = 0 is forbidden if ∆J = 0
...
13
Interaction with electromagnetic fields
The Hamiltonian of an electron in an electromagnetic field is given by:
H=
2
2
1
~ ·L
~ + e A2 − eV
~ 2 − eV = − ¯h ∇2 + e B
(~
p + eA)
2µ
2µ
2µ
2µ
where µ is the reduced mass of the system
...
strong fields or macroscopic motions
...
Because f = f (x, t), this is called a local gauge transformation, in contrast with a global gauge
transform which can always be applied
...
14
Perturbation theory
10
...
1
Time-independent perturbation theory
To solve the equation (H0 + λH1 )ψn = En ψn one has to find the eigenfunctions of H = H0 + λH1
...
Because φn is a complete set holds:
X
ψn = N (λ) φn +
cnk (λ)φk
k6=n
(1)
(2)
When cnk and En are being developed to λ: cnk = λcnk + λ2 cnk + · · ·
(1)
(2)
En = En0 + λEn + λ2 En + · · ·
51
Chapter 10: Quantum physics
(1)
and this is put into the Schr¨odinger equation the result is: En = hφn |H1 |φn i and
hφm |H1 |φn i
if m 6= n
...
So in first order holds: ψn = φn +
φk
...
In that case an orthonormal set eigenfunctions φni is chosen for each level n, so that hφmi |φnj i = δmn δij
...
Substitution in the Schr¨odinger equation and
Eni = Eni
+ λEni is approximated by Eni
P
P
(1)
taking dot product with φni gives:
αi hφnj |H1 |φni i = En αj
...
i
10
...
2
i
Time-dependent perturbation theory
∂ψ(t)
= (H0 + λV (t))ψ(t)
∂t
X
−iEn0 t
(1)
φn with cn (t) = δnk + λcn (t) + · · ·
and the development ψ(t) =
cn (t) exp
h
¯
n
From the Schr¨odinger equation i¯
h
follows:
c(1)
n (t)
λ
=
i¯
h
Zt
0
′
hφn |V (t )|φk i exp
10
...
15
...
For the total wavefunction of a system of identical indistinguishable particles holds:
1
...
r
...
interchange of the coordinates (spatial and spin) of each pair of particles
...
2
...
r
...
interchange of the
coordinates (spatial and spin) of each pair of particles
...
When a and b are
the quantum numbers of electron 1 and 2 holds:
ψS (1, 2) = ψa (1)ψb (2) + ψa (2)ψb (1) , ψA (1, 2) = ψa (1)ψb (2) − ψa (2)ψb (1)
Because the particles do not approach each other closely at ψA the repulsion energy in this state is
smaller
...
52
Equations in Physics by ir
...
C
...
Wevers
For N particles the symmetric spatial function is given by:
X
ψS (1,
...
N )
1
The antisymmetric wavefunction is given by the determinant ψA (1,
...
15
...
If the 2 atoms approach each other there are
two possibilities:
the total wavefunction approaches the bonding function with lower √
total energy
√
ψB = 12 2(φa + φb ) or approaches the anti-bonding function with higher energy ψAB = 21 2(φa − φb )
...
r
...
the connecting axis, like a combination of two s-orbitals it
is called a σ-orbital, otherwise a π-orbital, like the combination of two p-orbitals along two axes
...
hψ|ψi
The energy calculated with this method is always higher than the real energy if ψ is only an approximation for the solutions of Hψ = Eψ
...
Applying this on the function
ψ = ci φi one finds: (Hij − ESij )ci = 0
...
Here, Hij = hφi |H|φj i and Sij = hφi |φj i
...
Sii = 1 and Sij is the overlap integral
...
This results in a large electron density between the nuclei
and therefore a repulsion
...
In some atoms, like C, it is energetical more suitable to form orbitals which are a linear combination
of the s, p and d states
...
SP-hybridization: ψsp = 21 2(ψ2s ± ψ2pz )
...
Further the 2px and 2py orbitals remain
...
SP2 hybridization: ψsp2 = ψ2s +c1 ψ2pz +c2 ψ2py , where (c1 , c2 ) ∈ {(1, −1, 0), (1, 1, −1), (1, 1, 1)}
...
3
...
The 4 SP3 orbitals form a tetraheder
with the symmetry axes at an angle of 109◦ 28′
...
16
Quantum statistics
If a system exists in a state in which one has not the maximal amount of information about the
system, it can be described with a density matrix ρ
...
i
If ψ is developed to an orthonormal basis {φk } as: ψ
hAi =
X
(i)
=
P
k
(i)
ck φk ,
holds:
(Aρ)kk = Tr(Aρ)
k
P
where ρlk = c∗k cl
...
Further holds ρ =
ri |ψi ihψi |
...
For the time-dependence holds (in the Schr¨odinger image operators are not explicitly time-dependent):
i¯h
dρ
= [H, ρ]
dt
53
Chapter 10: Quantum physics
For a macroscopic system in equilibrium holds [H, ρ] = 0
...
For a mixed state of M orthonormal quantum
hHi =
pn En = −
∂kT
n
n
states with probability 1/M follows: S = k ln(M )
...
This function can be found by finding the maximum of the function
P which gives
nk = N and
the number of states with Stirling’s equation: ln(n!) ≈ n ln(n) − n, and the conditions
k
P
nk Wk = W
...
For indistinguishable particles who do not obey the exclusion principle the possible number of states is given by:
P = N!
Y g nk
k
k
nk !
This results in the Bose-Einstein statistics
...
They are given by:
gk
N
1
...
nk ∈ {0, 1}, nk =
Z
exp((E
−
µ)/kT ) + 1
g
k
P
with ln(Zg ) = gk ln[1 + exp((Ei − µ)/kT )]
...
Bose-Einstein statistics: half odd-integer spin
...
Here,
Zg is the large-canonical state sum and µ the chemical potential
...
N is the total number of particles
...
Fermi-Dirac statistics: nk =
gk
...
Bose-Einstein statistics: nk =
gk
...
1
Introduction
The degree of ionization α of a plasma is defined by: α =
ne
ne + n0
where ne is the electron density and n0 the density of the neutrals
...
The probability that a test particle has a collision with another is given by dP = nσdx where σ is the
cross section
...
The mean free path is given by λv = 1/nσ
...
The number of collisions per unit of time and volume
between particles of kind 1 and 2 is given by n1 n2 hσvi = Kn1 n2
...
Here, λD is the Debye length
...
Deviations of charge neutrality by thermic motion are
compensated by oscillations with frequency
s
ne e 2
ωpe =
me ε 0
The distance of closest approximation when two equal charged particles collide for a deviation of π/2
−1/3
is 2b0 = e2 /(4πε0 21 mv 2 )
...
Here Lp := |ne /∇ne | is the gradient length of the plasma
...
2
Transport
Relaxation times are defined as τ = 1/νc
...
Because
for e-i collisions the energy
p transfer is only ∼ 2me /mi this is a slow process
...
The relaxation for e-o interaction is much more complicated
...
54
55
Chapter 11: Plasma physics
The resistivity η = E/J of a plasma is given by:
√
e2 me ln(ΛC )
ne e 2
√
=
η=
me νei
6π 3ε20 (kTe )3/2
The diffusion coefficient D is defined via the flux Γ by ~Γ = n~vdiff = −D∇n
...
One finds that D = 31 λv v
...
For magnetized plasma’s λv must be replaced with the cyclotron radius
...
electrical fields also holds J~ = neµE
The Einstein ratio is:
D
kT
=
µ
e
Because a plasma is electrical neutral electrons and ions are strongly coupled and they don’t diffuse
independent
...
From this follows that
kTe µi
kTe /e − kTi /e
≈
Damb =
1/µe − 1/µi
e
In an external magnetic field B0 particles will move in spiral orbits with cyclotron radius ρ = mv/eB0
and with cyclotron frequency Ω = B0 e/m
...
A plasma
is called magnetized if λv > ρe,i
...
In case of magnetic confinement holds: ∇p = J~ × B
...
For a uniform B-field holds: p = nkT = B 2 /2µ0
...
2
~ = Er ~er + Ez ~ez and B
~ = Bz ~ez the E
~ ×B
~ drift results in a velocity ~u = (E
~ × B)/B
~
If E
and the
˙
velocity in the r, ϕ plane is r(r,
˙ ϕ, t) = ~u + ρ
~(t)
...
3
Elastic collisions
11
...
1
General
The scattering angle of a particle in interaction with
another particle, as shown in the figure at the right is:
χ = π − 2b
Z∞
ra
b
@
I
@
R
dr
r2
s
1−
W (r)
b2
−
r2
E0
Particles with an impact parameter between b and b +
db, going through a ring with dσ = 2πbdb leave the
scattering area at a solid angle dΩ = 2π sin(χ)dχ
...
ra
b 6
?
ϕ
χ
M
For low energies, O(1 eV), σ has a Ramsauer minimum
...
I(Ω) for angles 0 < χ < λ/4 is larger than the classical value
...
J
...
A
...
3
...
This gives b = b0 cot( 12 χ)
and
b20
b ∂b
=
I(Ω =
sin(χ) ∂χ
4 sin2 ( 12 χ)
Because the influence of a particle vanishes at r = λD holds: σ = π(λ2D − b20 )
...
For this quantity holds: ΛC = λD /b0 = 9n(λD )
...
3
...
Repulsing nuclear forces prevent this to happen
...
The cross section for capture σorb = πb2a is
called the Langevin limit, and is a lowest estimate for the total cross section
...
3
...
3
...
The scattering is free of collective
effects if kλD ≪ 1
...
If relativistic effects become important,
this limit of Compton scattering (which is given by λ′ − λ = λC (1 − cos χ) with λC = h/mc) can not
be used any more
...
4
Thermodynamic equilibrium and reversibility
For a plasma in equilibrium holds Planck’s radiation law and the Maxwellian velocity distribution:
8πhν 3
E
1
2πn √
ρ(ν, T )dν =
E exp −
dE
dν , N (E, T )dE =
c3 exp(hν/kT ) − 1
kT
(πkT )3/2
“Detailed balancing” means that the number of reactions in one direction equals the number of
reactions in the opposite direction because both processes have equal probability if one corrects for
the used phase space
...
For electrons holds g = 2, for excited
states usually holds g = 2j + 1 = 2n2
...
This factor causes the Saha-jump
...
5
Inelastic collisions
11
...
1
Types of collisions
gp
∆Epq
K(p, q, T ) exp
gq
kT
The kinetic energy can be split in a part of and a part in the center of mass system
...
This energy is given by
E=
m1 m2 (v1 − v2 )2
2(m1 + m2 )
Some types of inelastic collisions important for plasma physics are:
→ Aq + e−
1
...
Decay: Aq ←
58
Equations in Physics by ir
...
C
...
Wevers
3
...
radiative recombination: A+ + e− →
← Ap + hf
5
...
Associative ionisation: A∗∗ + B ←
→ Ar+ + Ne + e−
7
...
v
...
Charge transfer: A+ + B ←
→ A + A+
9
...
5
...
This results in
dσ
πZ 2 e4
=
d(∆E)
(4πε0 )2 E(∆E)2
Then follows for the transition p → q: σpq (E) =
πZ 2 e4 ∆Eq,q+1
(4πε0 )2 E(∆E)2pq
For ionisation from state p holds in good approximation: σp =
For resonant charge transfer holds: σex =
11
...
The intensity I af
a line is given by Ipq = hf Apq np /4π
...
Because
the states have a finite life time
...
From the uncertaincy relation then follows: ∆(hν) · τp = 21 h
q
∆ν =
1
=
4πτp
P
Apq
q
4π
The natural line width is usually ≪ than the broadening from the following two mechanisms:
59
Chapter 11: Plasma physics
2
...
3
...
The natural broadening and the Stark broadening result in a Lorentz profile of a spectral line: kν =
1
1
2
2
2 k0 ∆νL /[( 2 ∆νL ) + (ν − ν0 ) ]
...
The number of transitions p → q is given by np Bpq ρ and by np nhf hσa ci = np (ρdν/hν)σa c where dν
is the line width
...
The background radiation in a plasma originates from two processes:
1
...
The emission is given by:
hc
C1 zi ni ne
1 − exp −
ξf b (λ, Te )
εf b = 2 √
λ
λkTe
kTe
with C1 = 1, 63 · 10−43 Wm4 K1/2 sr−1 and ξ the Biberman factor
...
Free-free radiation, originating from the acceleration of particles in the EM-field of other particles:
hc
C1 zi ni ne
ξf f (λ, Te )
exp −
εf f = 2 √
λ
λkTe
kTe
11
...
This is also true in magnetic fields because ∂ai /∂xi = 0
...
~
The
total
density
is
given
by
n
=
F
d~
v
t
R
and ~v F d~v = nw
...
Mass balance: (BTE)d~v ⇒
∂t
∂t bs
Z
dw
~
~
2
...
Energy balance: (BTE)mv 2 d~v ⇒
2 dt
2
60
Equations in Physics by ir
...
C
...
Wevers
~ +w
~ is the average acceleration, ~q = 1 nm ~v 2~vt the heat flow, Q =
Here, h~a i = e/m(E
~ × B)
t
2
Z
mvt2 ∂F
~ is a friction term and p = nkT the
d~v the source term for energy production, R
r
∂t bs
pressure
...
11
...
solution is valid, where ∀p>1 [(∂np /∂t = 0) ∧ (∇ · (np w
~ p ) = 0)]
...
Further holds for all collision-dominated levels that
p p
−x
δbp := bp − 1 = b0 peff with peff = Ry/Epi and 5 ≤ x ≤ 6
...
Even in
plasma’s far from equilibrium the excited levels will eventually reach ESP, so from a certain level up
the level densities can be calculated
...
recomb
...
recomb
coll
...
ne np
coll
...
X
Kpq + nenp
q
11
...
deexcit
...
deex
...
excit
...
ion
...
deex
...
For
electromagnetic waves with complex wave number k = ω(n + iκ)/c in one dimension one finds:
Ex = E0 e−κωx/c cos[ω(t − nx/c)]
...
The dielectric tensor E, with property:
~k · (~E~ · E)
~ =0
~ ~
is given by E~ = I~ − ~
~σ /iε0 ω
...
~er is connected with a right rotating field for
which iEx /Ey = 1 and ~el is connected with a left rotating field for which iEx /Ey = −1
...
From this the following solutions can be obtained:
A
...
1
...
This describes a longitudinal linear polarized wave
...
n2 = L: a left, circular polarized wave
...
n2 = R: a right, circular polarized wave
...
θ = π/2: transmission ⊥ the B-field
...
n2 = P : the ordinary mode: Ex = Ey = 0
...
2
...
Resonance frequencies are frequencies for which n2 → ∞, so vf = 0
...
For R → ∞ this gives the electron cyclotron resonance frequency ω = Ωe , for L → ∞ the ion cyclotron
resonance frequency ω = Ωi and for S = 0 holds for the extraordinary mode:
mi Ω2i
m2i Ω2i
Ω2i
α2 1 −
=
1
−
1
−
me ω 2
m2e ω 2
ω2
Cut-off frequencies are frequencies for which n2 = 0, so vf → ∞
...
In the case that β 2 ≫ 1 one finds Alfv´en waves propagating parallel to the field lines
...
Chapter 12
Solid state physics
12
...
A lattice can be constructed from primitive cells
...
This periodicity is excellent to use Fourier analysis: n(~r) is developed as:
X
~ · ~r )
n(~r) =
nG exp(iG
G
with
nG =
1
Vcell
ZZZ
~ · ~r )dV
...
If G
~ is written as G
~ = v1~b1 + v2~b2 + v3~b3 with vi ∈ IN , follows for
G
~
the vectors bi , cyclical:
~bi = 2π ~ai+1 × ~ai+2
...
From this follows for parallel lattice planes (Bragg
if: ∆k = G with ∆~k = ~k − ~k ′
...
The Brillouin zone is defined as a Wigner-Seitz cell in the reciprocal lattice
...
2
Crystal binding
A distinction can be made between 4 kinds of binding:
1
...
Ion bond
3
...
Metalic bond
...
The potential energy for two parallel spins is higher than the potential energy for two
antiparallel spins
...
In
that case binding is not possible
...
3
Crystal vibrations
12
...
1
lattice with one kind of atoms
In this model for crystal vibrations, only nearest-neighbor interactions are taken into account
...
This gives: us = exp(iKsa)
...
This gives:
ω2 =
4C
sin2 ( 21 Ka)
M
Only vibrations with a wavelength within the first Brillouin Zone have a physical significance
...
The group velocity of these vibrations is given by:
dω
=
vg =
dK
r
Ca2
cos( 12 Ka)
...
Here, there is a standing wave
...
3
...
The upper line describes the optical
branch, the lower line the acoustical branch
...
This
results in a much larger induced dipole moment for optical
oscillations, and also a stronger emission and absorption of
radiation
...
12
...
3
q
q
0
2C
M2
2C
M1
- K
π/a
Phonons
The quantummechanical excitation of a crystal vibration with an energy h
¯ ω is called a phonon
...
Their total momentum is 0
...
Because
phonons have no spin they behave like bosons
...
3
...
J
...
A
...
The thermal heat capacity is then:
XZ
∂U
(¯
hω/kT )2 exp(¯
hω/kT )
dω
Crooster =
=k
D(ω)
∂T
(exp(¯
hω/kT ) − 1)2
λ
The dispersion relation in one dimension is given by:
D(ω)dω =
L dK
L dω
dω =
π dω
π vg
In three dimensions one applies periodic boundary conditions on a cube with N 3 primitive cells and
a volume L3 : exp(i(Kx x + Ky y + Kz z)) ≡ exp(i(Kx (x + L) + Ky (y + L) + Kz (z + L))
...
±
L
L
L
L
~ per volume (2π/L)3 in K-space, or:
So there is only one permitted value of K
3
V
L
=
2π
8π 3
~
~
permitted K-values
per unit volume in K-space,
for each polarization and each branch
...
The density of states for each polarization is, according to the Einstein model:
ZZ
dN
V K 2 dK
dAω
V
D(ω) =
=
=
dω
2π 2
dω
8π 3
vg
The Debye model for thermal heat capacities is a low-temperature approximation which is valid till
≈ 50K
...
From this follows: D(ω) = V ω 2 /2π 2 v 3 , where v is the
speed of sound
...
ex − 1
Here, xD = h
¯ ωD /kT = θD /T
...
Because xD → ∞ for T → 0 follows from this:
U = 9N kT
T
θD
3 Z∞
0
x3 dx
3π 4 N kT 4
∼ T4
=
ex − 1
5θD
and CV =
12π 4 N kT 3
∼ T3
3
5θD
In the Einstein model for the thermal heat capacity one considers only phonons at one frequency, an
approximation for optical phonons
...
4
Magnetic field in the solid state
12
...
1
Dielectrics
The quantummechanical origin of diamagnetism is the Larmorprecession of the spin of the electron
...
If the magnetic part of the force is not strong enough to significant
deform the orbit holds:
s
2
eB
eB
eB
Fc (r) eB
2
2
±
ω = ω0 ±
(ω0 + δ) ⇒ ω =
+ · · · ≈ ω0 ±
= ω0 ± ωL
ω0 ±
ω =
mr
m
m
2m
2m
Here, ωL is the Larmor frequency
...
So there
is a net circular current which results
in a magnetic
moment ~µ
...
If N is the number of atoms in the crystal
~ = ~µN :
follows for the susceptibility, with M
χ=
12
...
2
Paramagnetism
µ0 N Ze2
2
µ0 M
r
=−
B
6m
~ = mJ gµB B,
Starting with the splitting of energy levels in a weak magnetic field: ∆Um − ~µ · B
and
with
a
distribution
f
∼
exp(−∆U
/kT
),
one
finds
for
the
average
magnetic
hµi =
m
m
P
P
P
P
P moment
fm µ/ fm
...
12
...
3
Ferromagnetism
A ferromagnet behaves like a paramagnet above a critical temperature Tc
...
The treatment is further analogous with
~ E = λµ0 M
a field BE parallel with M is postulated: B
paramagnetism:
C
M
µ0 M = χp (Ba + BE ) = χp (Ba + λµ0 M ) = µ0 1 − λ
T
From this follows for a ferromagnet: χF =
µ0 M
C
=
this is Weiss-Curie’s law
...
This is clearly unrealistic and suggests
an other mechanism
...
J is an overlap integral given by:
between two neighbor atoms: U = −2J S
J = 3kTc /2zS(S + 1), with z the number of neighbors
...
J > 0: Si and Sj become parallel: the material is a ferromagnet
...
J < 0: Si and Sj become antiparallel: the material is an antiferromagnet
...
Starting with a model with only nearest
neighbor interaction one can write:
~p · (S
~p−1 + S
~p+1 ) ≈ ~µp · B
~p
U = −2J S
~p−1 + S
~p+1 )
~ p = −2J (S
with B
gµB
66
Equations in Physics by ir
...
C
...
Wevers
The equation of motion for the magnons becomes:
~
dS
2J ~
~p−1 + S
~p+1 )
Sp × (S
=
dt
¯
h
~p =
The treatment is further analogous with phonons: postulate traveling waves of the type S
~u exp(i(pka − ωt))
...
5
Free electron Fermi gas
12
...
1
Thermal heat capacity
The solution with period L of the one-dimensional Schr¨odinger equation is: ψn (x) = A sin(2πx/λn)
with nλn = 2L
...
The Fermi level is the
uppermost filled level in the ground state, which has the Fermi-energy EF
...
In 3 dimensions
holds:
2/3
2 1/3
3π N
¯h2 3π 2 N
kF =
and EF =
V
2m
V
3/2
V
2mE
The number of states with energy ≤ E is then: N =
...
and the density of states becomes: D(E) =
E=
2
2
dE
2π
2E
¯h
The heat capacity of the electrons is approximately 0
...
This
is caused by the Pauli exclusion principle and the Fermi-Dirac distribution: only electrons within an
energy range ∼ kT of the Fermi level are excited thermally
...
The internal energy then becomes:
U ≈ N kT
∂U
T
T
and C =
≈ Nk
TF
∂T
TF
A more accurate analysis gives: Celectrons = 21 π 2 N kT /TF ∼ T
...
12
...
2
Electric conductance
The equation of motion for the charge carriers is: F~ = md~v /dt = h
¯ d~k/dt
...
If τ is the characteristic collision time of the electrons, δ~k remains
by δ~k = ~k(t) − ~k(0) = −eEt/¯
~ with µ = eτ /m the mobility of the electrons
...
Then holds: h~v i = µE,
~ = E/ρ
~ = neµE
...
T →0
12
...
3
The Hall-effect
If a magnetic field is applied ⊥ at the direction of the current the charge carriers will be pushed aside
by the Lorentz force
...
If J~ = J~ex
~ = B~ez then Ey /Ex = µB
...
The Hall voltage is given by: VH = Bvb = IB/neh where b is the width of the
material and h de height
...
5
...
From this follows for the Wiedemann-Franz ratio: κ/σ = 31 (πk/e)2 T
...
6
Energy bands
In the tight-bond approximation it is assumed that ψ = eikna φ(x − na)
...
So this gives a cosine superimposed on the atomic
energy, which can often be approximated with a harmonic oscillator
...
This is a traveling wave
...
The
probability density |ψ(−)|2 is low near the atoms of the lattice and high between them
...
Suppose that U (x) = U cos(2πx/a), than the bandgap
is given by:
Z1
Egap = U (x) |ψ(+)|2 − |ψ(−)|2 dx = U
0
12
...
Here it is assumed that the momentum of the absorbed photon can be
neglected
...
conduction
E
band
6
•
6
ωg
◦
E
6
•
Ω
Direct transition
6
ω
◦
Indirect transition
This difference can also be observed in the absorption spectra:
absorption
absorption
6
6
hωg
¯
-E
Direct semiconductor
...
...
...
...
...
J
...
A
...
When light is absorbed holds: ~kh = −~ke , Eh (~kh ) = −Ee (~ke ), ~vh = ~ve and mh = −m∗e
if the conduction band and the valence band have the same structure
...
It is defined
by:
2 −1
d E
dp/dt
dK
F
2
∗
=h
¯
=
=h
¯
m =
a
dvg /dt
dvg
dk 2
with E = h
¯ ω and vg = dω/dk and p = h
¯ k
...
From this follows for the concentrations of
the holes p and the electrons n:
n=
Z∞
De (E)fe (E)dE = 2
Ec
For the product np follows: np = 4
kT
2π¯h2
3
m∗ kT
2π¯h2
3/2
exp
µ − Ec
kT
p
Eg
m∗e mh exp −
kT
In an intrinsic (no impurities) semiconductor holds: ni = pi , in a n − type holds: n > p and in a
p − type holds: n < p
...
The excitation
energy of an exciton is smaller than the bandgap because the energy of an exciton is lower than the
energy of a free electron and a free hole
...
12
...
8
...
The BCS-model predicts for the transition temperature
Tc :
−1
Tc = 1, 14ΘD exp
U D(EF )
while experiments find for Hc in approximation:
T2
Hc (T ) ≈ Hc (Tc ) 1 − 2
...
There are type I and type II superconductors
...
This holds for a type I
~ = µ0 M
ductor is a perfect diamagnet holds in the superconducting state: H
superconductor, for a type II superconductor this only holds to a certain value Hc1 , for higher values
of H the superconductor is in a vortex state to a value Hc2 , which can be 100 times Hc1
...
This is shown in the figures below
...
This
means that there is a twist in the T − S diagram and a discontinuity in the CX − T diagram
...
8
...
The electron
wavefunction in one superconductor is ψ1 , in the other ψ2
...
The
h
√
√
electron wavefunctions are written as √
ψ1 = n1 exp(iθ1 ) and ψ2 = n2 exp(iθ2 )
...
From this follows, if n1 ≈ n2 :
∂θ2
∂n2
∂n1
∂θ1
=
and
=−
∂t
∂t
∂t
∂t
The Josephson effect results in a current density through the insulator depending on the phase
difference as: J = J0 sin(θ2 − θ1 ) = J0 sin(δ), where J0 ∼ T
...
This gives: J = J0 sin θ2 − θ1 −
h
¯
i¯
h
So there is an oscillation with ω = 2eV /¯h
...
8
...
The size of a flux
s ∈ IN
...
12
...
4
Macroscopic quantum interference
From θ2 − θ1 = 2eΨ/¯
h follows for two parallel junctions: δb − δa =
2eΨ
, so
¯h
eΨ
This gives maxima if eΨ/¯h = sπ
...
8
...
J
...
A
...
ε0 mc2 /nq 2
...
Magnetic fields within
a superconductor drop exponentially
...
8
...
(There is till now no explanation for
high-Tc superconductance)
...
Because the
interaction with the lattice these pseudo-particles have a mutual attraction
...
It can be proved that this ground state is perfect
diamagnetic
...
Fluxquantization prevents transitions
between these states
...
A flux quantum is the equivalent of about 104 electrons
...
This is also very improbable
...
p
x3 dx
π4
=
ex + 1
15
Chapter 13
Theory of groups
13
...
1
...
∀A,B∈G ⇒ A • B ∈ G: G is closed
...
∀A,B,C∈G ⇒ (A • B) • C = A • (B • C): G the associative law
...
∃E∈G so that ∀A∈G A • E = E • A = A: G has a unit element
...
∀A∈G ∃A−1 ∈G z
...
d
...
If also holds:
5
...
13
...
2
The Cayley table
Each element arises only once in each row and column of the Cayley or multiplication table: because
EAi = A−1
k (Ak Ai ) = Ai each Ai appears once
...
13
...
3
Conjugated elements, subgroups and classes
B is conjugate with A if ∃X∈G such that B = XAX −1
...
If B and C are conjugate with A, B is also conjugate with C
...
A conjugacy class is the maximum collection of conjugated elements
...
Some theorems:
• All classes are completely disjoint
...
• E is the only class which is also a subgroup because all other classes have no unit element
...
The physical interpretation of classes: elements of a group are usually symmetry operations who map
a symmetrical object on itself
...
The
opposite need not to be true
...
J
...
A
...
1
...
The mapping from group G1
to G2 , so that the multiplication table remains the same is a homomorphic mapping
...
A representation is a homomorphic mapping of a group to a group of square matrices with the usual
matrix multiplication as the combining operation
...
The following holds:
Γ(E) = II , Γ(AB) = Γ(A)Γ(B) , Γ(A−1 ) = [Γ(A)]−1
For each group there are 3 possibilities for a representation:
1
...
2
...
3
...
An equivalent representation is obtained by performing an unitary base transformation: Γ′ (A) =
S −1 Γ(A)S
...
1
...
If this is not possible the representation is called irreducible
...
13
...
2
...
The opposite holds (of course) also
...
13
...
2
The fundamental orthogonality theorem
For a set of inequivalent, irreducible, unitary representations holds that, if h is the number of elements
in the group and ℓi is the dimension of the ith
¯ representation:
X
R∈G
13
...
3
(j)
Γ(i)∗
µν (R)Γαβ (R) =
h
δij δµα δνβ
ℓi
Character
The character of a representation is given by the trace of the matrix and is therefore invariant for
base transformations: χ(j) (R) = Tr(Γ(j) (R))
Also holds, with Nk the number of elements in a conjugacy class:
X
k
Theorem:
n
X
i=1
ℓ2i = h
χ(i)∗ (Ck )χ(j) (Ck )Nk = hδij
73
Chapter 13: Theory of groups
13
...
3
...
These
transformations are a group
...
This is considered an active rotation
...
PR is the symmetry group of the physical system
...
A degeneracy who is not the result of a symmetry
is called an accidental degeneracy
...
, ℓn
...
Each
n corresponds with an other energy level
...
A fixed
(n)
choice of Γ(n) (R) defines the base functions ψν
...
Particle in a periodical potential: the symmetry operation is a cyclic group: note the operator
describing one translation over one unit as A
...
, Ah = E}
...
For 0 ≤ p ≤ h− 1 follows:
Γ(p) (An ) = e2πipn/h
2π
2πp
mod
, so: PA ψp (x) = ψp (x − a) = e2πip/h ψp (x), this gives Bloch’s
If one defines: k = −
ah
a
theorem: ψk (x) = uk (x)eikx , with uk (x ± a) = uk (x)
...
3
...
The perturbed system
has H = H0 + V, and symmetry group G ⊂ G0
...
The representation then
usually becomes reducible: Γ(n) = Γ(n1 ) ⊕ Γ(n2 ) ⊕
...
ℓn1 = dim(Γ(n1 ) )
ℓn2 = dim(Γ(n2 ) )
ℓn
ℓn3 = dim(Γ(n3 ) )
Spectrum H0
Spectrum H
(n)
Theorem: The set of ℓn degenerated eigenfunctions ψν with energy En is a basis for an ℓn dimensional irreducible representation Γ(n) of the symmetry group
...
3
...
J
...
A
...
P
(aj)
(j)
F can also be expressed in base functions ϕ: F =
cajκ ϕκ
...
However, this does happen if cjaκ = cja
...
13
...
4
The direct product of representations
Consider a physical system existing of two subsystems
...
Basefunctions are ϕκ (~xi ), 1 ≤ κ ≤ ℓi
...
These define a space D ⊗ D(2)
...
3
...
r
...
the basis ϕκ ϕλ one uses a new basis ϕµ
These basefunctions lie in subspaces D(ak)
...
κλ
(j)
and the inverse transformation by: ϕ(i)
κ ϕλ =
X
ϕ(aκ)
(akµ|iκjλ)
µ
akµ
(i)
(j)
(ak)
The Clebsch-Gordan coefficients are dot products in essence: (iκjλ|akµ) := hϕk ϕλ |ϕµ
13
...
6
i
Symmetric transformations of operators, irreducible tensor operators
Observables (operators) transform as follows under symmetry transformations: A′ = PR APR−1
...
75
Chapter 13: Theory of groups
Also an operator can be decomposed into symmetry types: A =
P
jk
a(j)
κ
=
(j)
ak , with:
!
ℓj X (j)∗
Γκκ (R) (PR APR−1 )
h
R∈G
Theorem: Matrix elements Hij of the operator H who is invariant under ∀A∈G , are 0 between
states who transform according to non-equivalent irreducible unitary representations or according to
(i)
(i)
different rows of such a representation
...
For H = 1 this
becomes the previous theorem
...
Here one tries
(i)
to diagonalize H
...
Perturbation calculus can be applied independent within each category
...
13
...
7
The Wigner-Eckart theorem
(i)
(j)
(k)
Theorem: The matrix element hϕλ |Aκ |ψµ i can only be 6= 0 if Γ(j) ⊗ Γ(k) =
...
If
this is the case holds (if Γ(i) appears only once, otherwise one has to sum over a):
(i)
(k)
(i)
(j)
(k)
hϕλ |A(j)
i
κ |ψµ i = (iλ|jκkµ)hϕ kA kψ
This theorem can be used to determine selection rules: the probability on a dipole transition is given
by (~ǫ is the direction of polarization of the radiation):
PD =
8π 2 e2 f 3 |r12 |2
with r12 = hl2 m2 |~ǫ · ~r |l1 m1 i
3¯
hε0 c3
Further it can be used to determine intensity ratios: if there is only one value of a the ratio of
the matrix elements are the Clebsch-Gordan coefficients
...
However, the intensity ratios are also dependent on the occupation of
the atomic energy levels
...
4
Continuous groups
Continuous groups have h = ∞
...
g
...
13
...
1
The 3-dimensional translation group
For the translation of wavefunctions over a distance a holds: Pa ψ(x) = ψ(x − a)
...
+ a
dx
2
dx2
¯h ∂
, this can be written
Because the momentum operator in quantum mechanics is given by: px =
i ∂x
as:
ψ(x − a) = e−iapx /¯h ψ(x)
76
13
...
2
Equations in Physics by ir
...
C
...
Wevers
The 3-dimensional rotation group
This group is called SO(3) because a faithful representation can be constructed from orthogonal 3 × 3
matrices with a determinant of +1
...
−y
i
∂y
∂z
So in an arbitrary direction holds:
Rotations:
Translations:
Pα,~n = exp(−iα(~n · J~ )/¯h)
Pa,~n = exp(−ia(~n · p~ )/¯h)
Jx , Jy and Jz are called the generators of the 3-dim
...
translation group
...
Rotations are not generally interchangeable: consider a rotation around axis ~n in the xz-plane over
an angle α
...
13
...
3
Properties of continuous groups
The elements R(p1 ,
...
, pn
...
g
...
It is demanded that the multiplication and inverse of an element R depend
continuous of the de parameters of R
...
The notion conjugacy class for continuous groups is defined equally as for
discrete groups
...
Summation over all group elements is for continuous groups replaced with an integration
...
g
...
, pn ))g(R(p1 ,
...
R
R
Because of the properties of the Cayley table is demanded:
f (R)dR = f (SR)dR
...
Define new variables p′ by: SR(pi ) = R(p′i )
...
dV ′
dV ′
∂p′j
77
Chapter 13: Theory of groups
For the translation group holds: g(~a) = constant = g(~0) because g(a~n )d~a′ = g(~0)d~a and d~a′ = d~a
...
5
Z
R
G
dR < ∞
...
The parameter space is a collection points ϕ~n within a sphere with radius π
...
An other way to define parameters is via Eulers angle’s
...
The spherical angles of axis 3 w
...
t
...
Now a rotation around axis 3 remains
possible
...
The spherical angles of the z-axis w
...
t
...
then the rotation of a quantummechanical system is described with:
~
ψ → e−iαJz h¯ e−iβJy /¯h e−iγJz /¯h ψ
...
All irreducible representations of SO(3) can be constructed with the behaviour of the spherical harmonics Ylm (θ, ϕ) with −l ≤ m ≤ l and for a fixed l:
X
(l)
PR Ylm (θ, ϕ) =
Ylm′ (θ, ϕ)Dmm′ (R)
m′
D(l) is an irreducible representation of dimension 2l + 1
...
This expression holds for all
rotations over an angle α because the classes of SO(3) are rotations around the same angle around
an axis with an arbitrary orientation
...
This is contradictionary because the dimension of the representation is given by
χ′ (0)
...
space which is invariant under rotations
...
J
...
A
...
So here holds E → ±II
...
If only one state changes sign the observable quantities do change
...
13
...
6
...
, |j1 − j2 |
...
⊕ D(|j1 −j2 |)
The states can be characterized with quantum numbers in two ways: with j1 , m1 , j2 , m2 and with
j1 , j2 , J, M
...
6
...
Suppose j = 0: this gives the identical representation with ℓj = 1
...
Because PR A0 PR−1 = A0 this operator is invariant, e
...
the Hamiltonian
of a free atom
...
~ = (Ax , Ay , Az )
...
A vector operator: A
equally as the cartesian components of ~r by definition
...
r
...
PR ψ and PR φ:
The transformed
PR ψ|PR Ax PR−1 |PR φ = hψ|Ax |φi
...
According to the equation for
characters this means one can choose base operators who transform like Y1m (θ, ϕ)
...
A cartesian tensor of rank 2: Tij is a quantity which transforms under rotations like Ui Vj , where
~ and V
~ are vectors
...
The 9 components can be split in 3 invariant subspaces with
dimension 1 (D(0) ), 3 (D(1) ) and 5 (D(2) )
...
I
...
This transforms as the scalar U
II
...
These transform as the vector
~ × V~ , so as D(1)
...
The 5 independent components of the traceless, symmetric tensor S:
Sij = 12 (Tij + Tji ) − 31 δij Tr(T )
...
79
Chapter 13: Theory of groups
Selection rules for dipole transitions
Dipole operators transform as D(1) : for an electric dipole transfer is the operator e~r, for a magnetic
e ~
~
2m (L + 2S)
...
This means that J ′ ∈ {J + 1, J, |J − 1|}: J ′ = J or J ′ = J ± 1, except
J ′ = J = 0
...
~ This can also
field is determined by the projection of M
be understood from the Wigner-Eckart theorem: from this follows that the matrix elements from all
~ follows:
vector operators show a certain proportionality
...
7
Applications in particle physics
The physics of a system does not change after performing a transformation ψ ′ = eiδ ψ where δ is a
constant
...
~ and φ at the same E
~ and B:
~ gauge
There exist some freedom in the choice of the potentials A
~
~
transformations of the potentials do not change E and B (See chapter 2 and 10)
...
This is a local gauge transformation: the phase of the wavefunction changes different at each position
...
This is now stated as
The physics of the system does not change if A
guide principle: the “right of existence” of the electromagnetic field is to allow local gauge invariance
...
The split-off
of charge in the exponent is essential: it allows one gauge field for all charged particles, independent
of their charge
...
The group elements now are PR =
exp(−iQΘ)
...
The weak interaction
together with the electromagnetic interaction can be described with a force field that transforms
according to U(1)⊗SU(2), and consists of the photon and three intermediar vector bosons
...
~ where Θn are real constants and Tn
In general the group elements are given by PR = exp(−iT~ · Θ),
P
operators (generators), like Q
...
The cijk
k
are the structure constants of the group
...
These constants can be found with the help of group product elements: because G is closed holds:
~ ~ ~′ ~
~ ~
~′ ~
~ ′′ ~
eiΘ·T eiΘ ·T e−iΘ·T e−iΘ ·T = e−iΘ ·T
...
The group SU(2) has 3 free parameters: because it is unitary there are 4 real conditions on 4 complex
parameters, and the determinant has to be +1, remaining 3 free parameters
...
Here, H is a Hermitian matrix
...
For each matrix of SU(2) holds that Tr(H)=0
...
J
...
A
...
So these matrices are a choice for the operators
~
of SU(2)
...
Abstractly, one can consider an isomorphic group where only the commutation rules are considered
known about the operators Ti : [T1 , T2 ] = iT3 , etc
...
g
...
Elementary
particles can be classified in isospin-multiplets, this are the irreducible representations of SU(2)
...
The isospin-singlet ≡ the identical representation: e−iT ·Θ = 1 ⇒ Ti = 0
2
...
The group SU(3) has 8 free parameters
...
The
Hermitian, traceless operators are 3 SU(2)-subgroups in the ~e1~e2 , ~e1~e3 and the ~e2~e3 plane
...
By taking a linear combination one gets 8 matrices
...
Chapter 14
Nuclear physics
14
...
The top is at
56
26 Fe, the most stable nucleus
...
a1 : Binding energy of the strong nuclear force, approximately ∼ A
...
a2 : Surface correction: the nucleons near the surface are less bound
...
a3 : Coulomb repulsion between the protons
...
a4 : Asymmetry term: a surplus of protons or neutrons has a lower binding energy
...
a5 : Pair off effect: nuclei with an even number of protons or neutrons are more stable because
groups of two protons or neutrons have a lower energy
...
Z even, N odd: ǫ = 0, Z odd, N even: ǫ = 0
...
In the shell model of the nucleus one assumes that a nucleon moves in an average field of other
~ · S:
~ ∆Vls = 1 (2l + 1)¯
hω
...
Further, there is a contribution of the spin-orbit coupling ∼ L
2
1
So each level (n, l) niveau is split in two, with j = l ± 2 , where the state with j = l + 21 has the
lowest energy
...
is not electromagnetical
...
Because −l ≤ m ≤ l
81
82
Equations in Physics by ir
...
C
...
Wevers
h there are 2(2l + 1) substates who exist independent for protons and neutrons
...
This is the case if N or Z ∈ {2, 8, 20, 28, 50, 82, 126}
...
2
The shape of the nucleus
An nucleus is in first approximation spherical with a radius of R = R0 A1/3
...
If the nuclear radius is measured with the charge distribution one obtains
R0 ≈ 1, 2 · 10−15 m
...
l = 1 gives dipole vibrations, l = 2 quadrupole, with a2,0 = β cos γ and a2,±2 = 12 2β sin γ
where β is the deformation factor and γ the shape parameter
...
The parity of the electric moment is ΠE = (−1)l , of the magnetic moment
ΠM = (−1)l+1
...
~
L and M
2mp
2mp
where gS is the spin-gyromagnetic ratio
...
The z-components of the magnetic moment are given by ML,z = µN ml and MS,z =
~ =
gS µN mS
...
gI (e/2mp)I
...
3
Radioactive decay
The number of nuclei decaying is proportional with the number of nuclei: N˙ = −λN
...
The half life time follows from τ 21 λ = ln(2)
...
The probability that N nuclei decay within a time interval
is governed by a Poisson distribution:
λN e−λ
dt
N!
P
If aP
nucleus can decay to more final states holds: λ =
λi
...
There are 5 types of natural radioactive decay:
P (N )dt = N0
1
...
Because nucleons tend to order themselves in groups
of 2p+2n this can be considered as a tunneling of a He2+ nucleus through a potential barrier
...
2
...
Here a proton changes into a neutron or vice versa:
p+ → n0 + W+ → n0 + e+ + νe , and n0 → p+ + W− → p+ + e− + ν e
...
Electron capture: here, a proton in the nucleus captures an electron (usually from the K-shell)
...
Spontaneous fission: a nucleus breaks apart
...
γ-decay: here the nucleus emits a high-energetic photon
...
Usually
the decay constant of electric multipole moments is larger than the one of magnetic multipole
moments
...
The parity of the emitted radiation
p is Π = Π · Π
...
14
...
4
...
From this follows that the intensity
of the beam decreases as −dI = Inσdx
...
R(θ, ϕ)
dσ
=
Because dR = R(θ, ϕ)dΩ/4π = Inxdσ follows:
dΩ
4πnxI
∆N
dσ
If N particles are scattered in a material with density n holds:
= n ∆Ω∆x
N
dΩ
1
Z1 Z2 e2
dσ
=
For Coulomb collisions holds:
dΩ C
8πε0 µv02 sin4 ( 12 θ)
14
...
2
Quantummechanical model for n-p scattering
The initial state is a beam of neutrons moving among the z-axis with wavefunction ψinit = eikz
and current density Jinit = v|ψinit |2 = v
...
The total wavefunction is then given by
ψ = ψin + ψscat = eikz + f (θ)
eikr
r
The particle flux of the scattered particles is v|ψscat |2 = v|f (θ)|2 dΩ
...
The wavefunction of the incoming particles can be expressed in a sum of angular momentum
wavefunctions:
X
ψinit = eikz =
ψl
l
The impactparameter is related to the angular momentum with L = bp = b¯hk, so bk ≈ l
...
For higher energies holds: σ =
4π X 2
sin (δl )
k2
l
84
14
...
3
Equations in Physics by ir
...
C
...
Wevers
Conservation of energy and momentum in nuclear reactions
If a particle P1 collides with a particle P2 which is in rest w
...
t
...
k>2
The minimal required kinetic energy T of P1 in the laboratory system to initialize the reaction is
P
m1 + m2 + mk
T = −Q
2m2
If Q < 0 there is a threshold energy
...
5
Radiation dosimetry
Radiometric quantities determine the strength of the radiation source(es)
...
Parameters describing a relation between
those are called interaction parameters
...
The deceleration of a heavy particle is described with the BetheBloch equation:
dE
q2
∼ 2
ds
v
The fluention is given by Φ = dN/dA
...
The energy loss is defined
by Ψ = dW/dA, and the energy flux density ψ = dΨ/dt
...
The mass absorption coefficient is given by µ/̺
...
An old unit is the R¨
ontgen: 1Ro= 2, 58 · 10−4 C/kg
...
The absorbed dose D is given by D = dEabs /dm, with unit Gy=J/kg
...
The dose tempo is defined as D
...
The equivalent dose H is an weight average of the absorbed dose per type of radiation, where for each
type radiation the effects on biological material is used for the weigh factor
...
Their unit is Sv
...
PIf the absorption is not equally distributed
also weigh factors w per organ needs to be used: H = wk Hk
...
1
Creation and annihilation operators
A state with more particles can be described with a collection occupation numbers |n1 n2 n3 · · ·i
...
This is a complete description because the particles are
indistinguishable
...
, and hΨ(t)|Ψ(t)i =
|cni (t)|2 = 1
...
H0 is the Hamiltonian for free particles and keeps |cni (t)|2 constant, Hint is
the interaction Hamiltonian and can increase or decrease one c2 at the cost of others
...
a is
the annihilation operator and a† the creation operator, and:
√
a(~ki )|n1 n2 · · · ni · · ·i =
ni |n1 n2 · · · ni − 1 · · ·i
√
† ~
ni + 1 |n1 n2 · · · ni + 1 · · ·i
a (ki )|n1 n2 · · · ni · · ·i =
Because the states are normalized holds a|0i = 0 and a(~ki )a† (~ki )|ni i = ni |ni i
...
The following commutation rules can be derived:
[a(~ki ), a(~kj )] = 0 ,
[a† (~ki ), a† (~kj )] = 0 ,
P
hωki
So for free spin-0 particles holds: H0 = a† (~ki )a(~ki )¯
[a(~ki ), a† (~kj )] = δij
i
15
...
3
The Lagrangian is given
x
...
J
...
A
...
With this, the Hamilton density becomes H(x) = Πα Φ
Quantization of a classical field is analogous to quantization in point mass mechanics: the field
functions are considered as operators obeying certain commutation rules:
[Φα (~x), Φβ (~x′ )] = 0 ,
15
...
Schr¨odinger picture: time-dependent states, time-independent operators
...
Heisenberg picture: time-independent states, time-dependent operators
...
Interaction picture: time-dependent states, time-dependent operators
...
From this follows:
i
15
...
With the definition
k02 = ~k 2 + M 2 := ωk2 and the notation ~k · ~x − ik0 t := kx the general solution of this equation is given
by:
q
i X 1
1 X 1 ~ ikx
~ ikx + a† (~k)e−ikx
√
a(k)e + a† (~k)e−ikx ; Π(x) = √
Φ(x) = √
2 ωk −a(k)e
2ωk
V ~
V ~
k
k
The field operators contain a volume V , which is used as normalizing factor
...
In general holds that the term with e−ikx , the positive frequency part, is the creation part, and the
negative frequency part is the annihilation part
...
Because Φ has
only one component this can be interpreted as a field describing a particle with zero spin
...
This
is consistent with the previously found result [Φ(~x, t, Φ(~x′ , t)] = 0
...
So the equations obey the locality postulate
...
The energy operator is given
by:
Z
X
H = H(x)d3 x =
¯hωk a† (~k)a(~k)
~
k
15
...
Noether’s theorem connects a continuous symmetry of L and an additive conservation law
...
Then holds
∂
∂L
α
=0
f
∂xν ∂(∂ν Φα )
This is a continuity equation ⇒ conservation law
...
The above Lagrange density is invariant for a change in phase Φ → Φeiθ : a global gauge
transformation
...
Because
this quantity is 0 for real fields a complex field is needed to describe charged particles
...
15
...
A scalar field Φ has the
˜
property that if it obeys the Klein-Gordon equation also the rotated field Φ(x)
:= Φ(Λ−1 x) obeys it
...
These can be written as:
∂
∂
~
˜
Φ(x)
= Φ(x)e−i~n·L with Lµν = −i¯h xµ
− xν
∂xν
∂xµ
For µ ≤ 3, ν ≤ 3 this are rotations, for ν = 4, µ 6= 4 this are Lorentz transformations
...
This results
~
−1
i~
n·S
in the condition D γλ D = Λλµ γµ
...
So:
˜
ψ(x)
= e−i(S+L) ψ(x) = e−iJ ψ(x)
The solutions of the Dirac equation are then given by:
ψ(x) = ur± (~
p )e−i(~p·~x±Et)
88
Equations in Physics by ir
...
C
...
Wevers
Here, r is an indication for the direction of the spin, and ± is the sign of the energy
...
A Lorentz-invariant dot product is defined
by ab := a† γ4 b, where a := a† γ4 is a row spinor
...
15
...
g
...
g
...
The energy operator is given by
H=
X
p
~
Ep~
2
X
r=1
p)
c†r (~
p )cr (~
p ) − dr (~
p )d†r (~
To prevent that the energy of positrons is negative the operators must obey anti commutation rules
in stead of commutation rules:
[cr (~
p ), c†r′ (~
p )]+ = [dr (~
p ), d†r′ (~
p )]+ = δrr′ δpp′ , all other anti commutators are 0
...
It appears
to be not possible to create two electrons with the same momentum and spin
...
An other method to see this is the fact that {Nr+ (~
p )}2 = Nr+ (~
p ): the occupation operators
have only eigenvalues 0 and 1
...
The expression for the current density now becomes Jµ = −ieN (ψγµ ψ)
...
By
an interchange of two fermion operators add a − sign, by interchange of two boson operators
not
...
15
...
All other commutators are 0
...
Further holds:
[Aµ (x), Aν (x′ )] = iδµν D(x − x′ ) with D(y) = ∆(y)|m=0
In spite of the fact that A4 = iV is imaginary in the classical case, A4 is still defined to be hermitian
because otherwise the sign of the energy becomes incorrect
...
If the potentials satisfy the Lorentz gauge condition ∂µ Aµ = 0 the E and B operators derived
from these potentials will satisfy the Maxwell equations
...
There is now demanded that only those states are permitted for which holds
∂A+
µ
|Φi = 0
∂xµ
This results in:
∂Aµ
∂xµ
= 0
...
With a local gauge transformation one obtains
N3 (~k) = 0 and N4 (~k) = 0
...
These photons are also responsible for
the stationary Coulomb potential
...
9
Interacting fields and the S-matrix
The S(scattering)-matrix gives a relation between the initial and final states for an interaction:
|Φ(∞)i = S|Φ(−∞)i
...
J
...
A
...
The S-matrix is
then given by: Sij = hΦi |S|Φj i = hΦi |Φ(∞)i
...
Each term corresponds with a possible process
...
Only terms with the correct number of particles in the initial and final state contribute to
a matrixelement hΦi |S|Φj i
...
The expressions for S (n) contain time-ordened products of normal products
...
The appearing operators describe the minimal changes necessary to
change the initial state in the final state
...
Some time-ordened products are:
T {Φ(x)Φ(y)}
o
n
T ψα (x)ψβ (y)
T {Aµ (x)Aν (y)}
= N {Φ(x)Φ(y)} + 12 ∆F (x − y)
n
o
F
(x − y)
= N ψα (x)ψβ (y) − 21 Sαβ
F
= N {Aµ (x)Aν (y)} + 21 δµν Dµν
(x − y)
Here, S F (x) = (γµ ∂µ − M )∆F (x), DF (x) = ∆F (x)|m=0 and
Z ikx
1
e
d3 k
3
ω~k
(2π)
∆F (x) =
Z −ikx
e
1
d3 k
(2π)3
ω~k
if x0 > 0
if x0 < 0
The term 21 ∆F (x − y) is called the contraction of Φ(x) and Φ(y), and is the expectation value of the
time-ordened product in the vacuum state
...
The graphical representation of these processes
are called Feynman diagrams
...
The contraction functions can also be written as:
Z
Z
eikx
−2i
iγµ pµ − M 4
−2i
4
F
F
d k and S (x) = lim
d p
∆ (x) = lim
eipx 2
2
2
4
ǫ→0
ǫ→0 (2π)4
k + m − iǫ
(2π)
p + M 2 − iǫ
In the expressions for S (2) this gives rise to terms δ(p + k − p′ − k ′ )
...
However, virtual particles do not obey the relation between energy and
momentum
...
10
Divergences and renormalization
It turns out that higher order contribute infinitely much because only the sum p + k of the fourmomentum of the virtual particles is fixed
...
In the
x-representation this can be understood because the product of two functions containing δ-like singularities is not well defined
...
It is assumed that an electron, if there would not be an electromagnetical field, would
have a mass M0 and a charge e0 unequal to the observed mass M and charge e
...
So this gives, with M = M0 + ∆M :
Le−p (x) = −ψ(x)(γµ ∂µ + M0 )ψ(x) = −ψ(x)(γµ ∂µ + M )ψ(x) + ∆M ψ(x)ψ(x)
and Hint = ieN (ψγµ ψAµ ) − i∆eN (ψγµ ψAµ )
...
11
Classification of elementary particles
Elementary particles can be categorized as follows:
1
...
Baryons: these exists of 3 quarks or 3 antiquarks
...
Mesons: these exists of one quark and one antiquark
...
Leptons: e± , µ± , τ ± , νe , νµ , ντ , ν e , ν µ , ν τ
...
Field quanta: γ, W± , Z0 , gluons, gravitons (?)
...
u
d
s
c
b
t
e+
µ+
τ+
νe
νµ
ντ
Where B is the baryon number and L the lepton number
...
T is the isospin, with T3 the projection of the isospin on the third
axis, C the charmness, S the strangeness and B∗ the bottomness
...
The hadrons are as follows composed
from (anti)quarks:
π0
π+
π−
K0
K0
K+
K−
D+
D−
D0
D0
F+
F−
1
2
√
2(uu+dd)
ud
du
sd
ds
us
su
cd
dc
cu
uc
cs
sc
J/Ψ
T
p+
p−
n0
n0
Λ
Λ
Σ+
Σ−
Σ0
Σ0
Σ−
cc
bb
uud
uud
udd
udd
uds
uds
uus
uus
uds
uds
dds
Σ+
Ξ0
0
Ξ
Ξ−
Ξ+
Ω−
Ω+
Λ+
c
∆2−
∆2+
∆+
∆0
∆−
dds
uss
uss
dss
dss
sss
sss
udc
uuu
uuu
uud
udd
ddd
Each quark can exist is two spin states
...
The quantum numbers are subject to conservation laws
...
Geometrical conservation laws are invariant under Lorentz transformations and the CPT-operation
...
J
...
A
...
Mass/energy because the laws of nature are invariant for translations in time
...
Momentum because the laws of nature are invariant for translations in space
...
Angular momentum because the laws of nature are invariant for rotations
...
These are:
1
...
2
...
3
...
4
...
5
...
6
...
The elementary particles can be classified into three families:
leptons
quarks
antileptons
antiquarks
1st generation
−
e
νe
d
u
+
e
νe
d
u
2nd generation
µ−
νµ
s
c
µ+
νµ
s
c
3rd generation
τ−
ντ
b
t
τ+
ντ
b
t
Quarks exist in three colors but because they are confined these colors can not be seen directly
...
The potential energy shall be high enough to create a
quark-antiquark pair when it is tried to free an (anti)quark from a hadron
...
15
...
Some processes which violate P symmetry but conserve the combination CP are:
1
...
Left-handed electrons appear more than 1000× as much as
right-handed ones
...
β-decay of spin-polarized 60 Co: 60 Co →60 Ni + e− + ν e
...
3
...
The CP-symmetry was found to be violated at the decay of neutral Kaons
...
The following holds: C|K0 i = η|K0 i
where η is a phase factor
...
From this follows that K0 and K0 are not eigenvalues of CP: CP|K0 i = |K0 i
...
A base of K01 and K02 is practical while
describing weak interactions
...
The expansion postulate must be used for weak decays:
|K0 i = 12 (hK01 |K0 i + hK02 |K0 i)
93
Chapter 15: Quantum field theory & Particle physics
The probability
to find a final state with CP= −1 is 12 | K02 |K0 |2 , the probability on CP=+1 decay
is 21 | K01 |K0 |2
...
15
...
g is the applying coupling constant
...
15
...
1
The electroweak theory
The electroweak interaction arises from the necessity to keep the Lagrange density invariant for
local gauge transformations of the group SU(2)⊗U(1)
...
If a fifth Dirac matrix is defined by:
0 0 1 0
0 0 0 1
γ5 := γ1 γ2 γ3 γ4 = −
1 0 0 0
0 1 0 0
the left and right handed solutions of the Dirac equation for neutrino’s are given by:
ψL = 12 (1 + γ5 )ψ
and ψR = 12 (1 − γ5 )ψ
It appears that neutrino’s are always left-handed while antineutrino’s are always right-handed
...
The group U(1)Y ⊗SU(2)T is taken as symmetry group for the electroweak interaction
because the generators of this group commute
...
J
...
A
...
The total Lagrange density (minus the fieldterms) for a scalar field now becomes:
′
g~
ψνe,L
µ
1 g
1
L0,EZ = −(ψνe,L , ψeL )γ ∂µ − i Aµ · ( 2 ~σ ) − 2 i Bµ · (−1)
−
ψeL
¯h
¯h
g′
ψeR γ µ ∂µ − 21 i (−2)Bµ ψeR
h
¯
Here, 21 ~σ are the generators of T and −1 and −2 the generators of Y
...
13
...
Their mass is probably generated by spontaneous
symmetry breaking
...
It is assumed that there exist an isospin-doublet of
scalar fields Φ with electrical charges +1 and 0
...
The
extra terms in L arising from these fields are globally U(1)⊗SU(2) symmetrical
...
The state on which
the system fluctuates is chosen so that
+
0
hΦ
i
=
hΦi =
hvi
Φ0
Because of this expectation value 6= 0 the SU(2) symmetry is broken but the U(1) symmetry is not
...
For this angle holds: sin2 (θW ) = 0, 255 ± 0, 010
...
according
to the weak theory this should be: MW = 83, 0 ± 0, 24 GeV/c2 and MZ = 93, 8 ± 2, 0 GeV/c2
...
13
...
A distinction can be made between two kinds of particles:
1
...
2
...
There exist three colors and three
anticolors
...
Because
left- and right handed quarks now belong to the same multiplet a mass term can be introduced
...
Chapter 15: Quantum field theory & Particle physics
15
...
Then,
Z
iS[x]
′ ′
F (x , t , x, t) = exp
d[x]
¯h
R
where S[x] is an action-integral: S[x] = L(x, x,
˙ t)dt
...
To each path is assigned a probability amplitude exp(iS/¯h)
...
In quantumvfieldtheory, the probability of the transition of a fieldoperator
Φ(~x, −∞) to Φ′ (~x, ∞) is given by
Z
iS[Φ]
′
d[Φ]
F (Φ (~x, ∞), Φ(~x, −∞)) = exp
¯h
with the action-integral
S[Φ] =
Z
Ω
L(Φ, ∂ν Φ)d4 x
Chapter 16
Astrophysics
16
...
The parallax is the angular
difference between two measurements of the position of the object from different views
...
The clusterparallax is used to determine the distance of a group of stars by using
their motion w
...
t a fixed background
...
This results, with vt = vr tan(θ), in:
R
R=
′′
vr tan(θ)
ˆ= 1
⇒ R
ω
p
where p is the parallax in arch seconds
...
A method to determine the distance of objects which are
somewhat further away, like galaxies and star clusters, uses the period-Brightness relation between
Cepheids
...
16
...
Earth receives s0 = 1, 374 kW/m2 from
the Sun
...
It is also given by:
L⊙ =
2
4πR⊙
Z∞
πFν dν
0
where πFν is the monochromatic radiation flux
...
If the fraction of the flux which reaches Earth’s surface is
called Aν , the transmission factor is given by Rν and the surface of the detector is given by πa2 , the
apparent brightness b is given by:
Z∞
2
fν Aν Rν dν
b = πa
0
The magnitude m is defined by:
b1
1
= (100) 5 (m2 −m1 ) = (2, 512)m2 −m1
b2
96
97
Chapter 16: Astrophysics
because the human eye perceives lightintensities logaritmical
...
The apparent brightness of a star if this star would be
at a distance of 10 pc is called the absolute brightness B: B/b = (ˆ
r /10)2
...
When an interstellar absorption
of 10−4 /pc is taken into account one finds:
M = (m − 4 · 10−4 rˆ) + 5 − 5 ·10 log(ˆ
r)
If a detector detects all radiation emitted by a source one would measure the absolute bolometric
magnitude
...
Further holds
L
10
Mb = −2, 5 · log
+ 4, 72
L⊙
16
...
When there is no absorption is the quantity Iν
independent of the distance to the source
...
ds is the thickness of
the layer
...
The layer is optically thin if
τν ≪ 1, the layer is optically thick as τν ≫ 1
...
Then also holds:
Iν (s) = Iν (0)e−τν + Bν (T )(1 − e−τν )
16
...
J
...
A
...
Further holds:
κ(r) = f (̺(r), T (r), composition) and ε(r) = g(̺(r), T (r), composition)
Convection will occur when the star meets the Schwartzschild criterium:
dT
dT
<
dr conv
dr stral
Otherwise the energy transfer shall be by radiation
...
For pp-chains holds µ ≈ 5 and for the CNO chaines
holds µ = 12 tot 18
...
Further holds:
8
L ∼ R4 ∼ Teff
...
5
log(L) = 8 ·10 log(Teff ) + constant
Energy production in stars
The net reaction from which most stars gain their energy is: 41 H →4 He + 2e+ + 2νe + γ
...
Two reaction chains are responsible for this reaction
...
The energy between brackets is the energy cried away
by the neutrino
...
The proton-proton chain can be divided in two subchains:
1
H + p+ →2 D + e+ + νe , and then 2 D + p →3 He + γ
...
pp1: 3 He +3 He → 2p+ +4 He
...
II
...
7 Be + e− → 7 Li + ν, dan 7 Li + p+ → 24 He + γ
...
ii
...
19,5 + (7,2) MeV
...
2
...
The first chain releases 25,03 + (1,69) MeV, the second 24,74 + (1,98) MeV
...
ր
15
O + e+ →
↑
14
N + p+ →
տ
→
15
N+ν
15
O+γ
←
−→
N + p+ → α +12 C
↓
12
C + p+ →13 N + γ
↓
13
N → 13 C + e+ + ν
↓
13
C + p+ →14 N + γ
←−
15
ց
ւ
15
N + p+ → 16 O + γ
↓
16
O + p+ → 17 F + γ
↓
17
F → 17 O + e+ + ν
↓
17
O + p+ → α +14 N