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130_notes.dvi

(Frankie) #1

  • 1 Course Summary

    • 1.1 Problems with Classical Physics

    • 1.2 Thought Experiments on Diffraction

    • 1.3 Probability Amplitudes

    • 1.4 Wave Packets and Uncertainty

    • 1.5 Operators

    • 1.6 Expectation Values

    • 1.7 Commutators

    • 1.8 The Schr ̈odinger Equation

    • 1.9 Eigenfunctions, Eigenvalues and Vector Spaces

    • 1.10 A Particle in a Box

    • 1.11 Piecewise Constant Potentials in One Dimension

    • 1.12 The Harmonic Oscillator in One Dimension

    • 1.13 Delta Function Potentials in One Dimension

    • 1.14 Harmonic Oscillator Solution with Operators

    • 1.15 More Fun with Operators

    • 1.16 Two Particles in 3 Dimensions

    • 1.17 Identical Particles

    • 1.18 Some 3D Problems Separable in Cartesian Coordinates

    • 1.19 Angular Momentum

    • 1.20 Solutions to the Radial Equation for Constant Potentials

    • 1.21 Hydrogen

    • 1.22 Solution of the 3D HO Problem in Spherical Coordinates

    • 1.23 Matrix Representation of Operators and States

    • 1.24 A Study ofℓ= 1 Operators and Eigenfunctions

    • 1.25 Spin 1/2 and other 2 State Systems

    • 1.26 Quantum Mechanics in an Electromagnetic Field

    • 1.27 Local Phase Symmetry in Quantum Mechanics and the Gauge Symmetry

    • 1.28 Addition of Angular Momentum

    • 1.29 Time Independent Perturbation Theory

    • 1.30 The Fine Structure of Hydrogen

    • 1.31 Hyperfine Structure

    • 1.32 The Helium Atom

    • 1.33 Atomic Physics

    • 1.34 Molecules

    • 1.35 Time Dependent Perturbation Theory

    • 1.36 Radiation in Atoms

    • 1.37 Classical Field Theory

    • 1.38 The Classical Electromagnetic Field

    • 1.39 Quantization of the EM Field

    • 1.40 Scattering of Photons

    • 1.41 Electron Self Energy

    • 1.42 The Dirac Equation

    • 1.43 The Dirac Equation



  • 2 The Problems with Classical Physics

    • 2.1 Black Body Radiation*

    • 2.2 The Photoelectric Effect

    • 2.3 The Rutherford Atom*

    • 2.4 Atomic Spectra*.

      • 2.4.1 The Bohr Atom*



    • 2.5 Derivations and Computations

      • 2.5.1 Black Body Radiation Formulas*

      • 2.5.2 The Fine Structure Constant and the Coulomb Potential



    • 2.6 Examples

      • 2.6.1 The Solar Temperature*

      • 2.6.2 Black Body Radiation from the Early Universe*.

      • 2.6.3 Compton Scattering*

      • 2.6.4 Rutherford’s Nuclear Size*.



    • 2.7 Sample Test Problems



  • 3 Diffraction

    • 3.1 Diffraction from Two Slits

    • 3.2 Single Slit Diffraction

    • 3.3 Diffraction from Crystals

    • 3.4 The DeBroglie Wavelength

      • 3.4.1 Computing DeBroglie Wavelengths



    • 3.5 Wave Particle Duality (Thought Experiments)

    • 3.6 Examples

      • 3.6.1 Intensity Distribution for Two Slit Diffraction*

      • 3.6.2 Intensity Distribution for Single Slit Diffraction*.



    • 3.7 Sample Test Problems



  • 4 The Solution: Probability Amplitudes

    • 4.1 Derivations and Computations

      • 4.1.1 Review of Complex Numbers

      • 4.1.2 Review of Traveling Waves



    • 4.2 Sample Test Problems



  • 5 Wave Packets

    • 5.1 Building a Localized Single-Particle Wave Packet

    • 5.2 Two Examples of Localized Wave Packets

    • 5.3 The Heisenberg Uncertainty Principle

    • 5.4 Position Space and Momentum Space

    • 5.5 Time Development of a Gaussian Wave Packet*

    • 5.6 Derivations and Computations

      • 5.6.1 Fourier Series*.

      • 5.6.2 Fourier Transform*

      • 5.6.3 Integral of Gaussian

      • 5.6.4 Fourier Transform of Gaussian*

      • 5.6.5 Time Dependence of a Gaussian Wave Packet*.

      • 5.6.6 Numbers

      • 5.6.7 The Dirac Delta Function



    • 5.7 Examples

      • 5.7.1 The Square Wave Packet

      • 5.7.2 The Gaussian Wave Packet*

      • 5.7.3 The Dirac Delta Function Wave Packet*

      • 5.7.4 Can I “See” inside an Atom

      • 5.7.5 Can I “See” inside a Nucleus

      • 5.7.6 Estimate the Hydrogen Ground State Energy



    • 5.8 Sample Test Problems



  • 6 Operators

    • 6.1 Operators in Position Space

      • 6.1.1 The Momentum Operator

      • 6.1.2 The Energy Operator

      • 6.1.3 The Position Operator

      • 6.1.4 The Hamiltonian Operator



    • 6.2 Operators in Momentum Space

    • 6.3 Expectation Values

    • 6.4 Dirac Bra-ket Notation

    • 6.5 Commutators

    • 6.6 Derivations and Computations

      • 6.6.1 Verify Momentum Operator

      • 6.6.2 Verify Energy Operator



    • 6.7 Examples

      • 6.7.1 Expectation Value of Momentum in a Given State

      • 6.7.2 Commutator ofEandt

      • 6.7.3 Commutator ofEandx.

      • 6.7.4 Commutator ofpandxn

      • 6.7.5 Commutator ofLxandLy



    • 6.8 Sample Test Problems



  • 7 The Schr ̈odinger Equation

    • 7.1 Deriving the Equation from Operators

    • 7.2 The Flux of Probability*

    • 7.3 The Schr ̈odinger Wave Equation

    • 7.4 The Time Independent Schr ̈odinger Equation

    • 7.5 Derivations and Computations

      • 7.5.1 Linear Operators

      • 7.5.2 Probability Conservation Equation*.



    • 7.6 Examples

      • 7.6.1 Solution to the Schr ̈odinger Equation in a Constant Potential



    • 7.7 Sample Test Problems



  • 8 Eigenfunctions, Eigenvalues and Vector Spaces

    • 8.1 Eigenvalue Equations

    • 8.2 Hermitian Conjugate of an Operator

    • 8.3 Hermitian Operators

    • 8.4 Eigenfunctions and Vector Space

    • 8.5 The Particle in a 1D Box

      • 8.5.1 The Same Problem with Parity Symmetry



    • 8.6 Momentum Eigenfunctions

    • 8.7 Derivations and Computations

      • 8.7.1 Eigenfunctions of Hermitian Operators are Orthogonal

      • 8.7.2 Continuity of Wavefunctions and Derivatives



    • 8.8 Examples

      • 8.8.1 Hermitian Conjugate of a Constant Operator

      • 8.8.2 Hermitian Conjugate of∂x∂



    • 8.9 Sample Test Problems



  • 9 One Dimensional Potentials

    • 9.1 Piecewise Constant Potentials in 1D

      • 9.1.1 The General Solution for a Constant Potential

      • 9.1.2 The Potential Step

      • 9.1.3 The Potential Well withE > 0 *.

      • 9.1.4 Bound States in a Potential Well*.

      • 9.1.5 The Potential Barrier



    • 9.2 The 1D Harmonic Oscillator

    • 9.3 The Delta Function Potential*.

    • 9.4 The Delta Function Model of a Molecule*.

    • 9.5 The Delta Function Model of a Crystal*

    • 9.6 The Quantum Rotor

    • 9.7 Derivations and Computations

      • 9.7.1 Probability Flux for the Potential Step*

      • 9.7.2 Scattering from a 1D Potential Well*

      • 9.7.3 Bound States of a 1D Potential Well*.

      • 9.7.4 Solving the HO Differential Equation*

      • 9.7.5 1D Model of a Molecule Derivation*.

      • 9.7.6 1D Model of a Crystal Derivation*



    • 9.8 Examples

    • 9.9 Sample Test Problems



  • 10 Harmonic Oscillator Solution using Operators

    • 10.1 IntroducingAandA†

    • 10.2 Commutators ofA,A†andH.

    • 10.3 Use Commutators to Derive HO Energies

      • 10.3.1 Raising and Lowering Constants



    • 10.4 Expectation Values ofpandx.

    • 10.5 The Wavefunction for the HO Ground State

    • 10.6 Examples

      • 10.6.1 The expectation value ofxin eigenstate

      • 10.6.2 The expectation value ofpin eigenstate

      • 10.6.3 The expectation value ofxin the state√^12 (u 0 +u 1 ).

      • 10.6.4 The expectation value of^12 mω^2 x^2 in eigenstate

        • 10.6.5 The expectation value of p

        • 2 min eigenstate



      • 10.6.6 Time Development Example



    • 10.7 Sample Test Problems



  • 11 More Fun with Operators

    • 11.1 Operators in a Vector Space

      • 11.1.1 Review of Operators

      • 11.1.2 Projection Operators|j〉〈j|and Completeness

      • 11.1.3 Unitary Operators



    • 11.2 A Complete Set of Mutually Commuting Operators

    • 11.3 Uncertainty Principle for Non-Commuting Operators

    • 11.4 Time Derivative of Expectation Values*.

    • 11.5 The Time Development Operator*.

    • 11.6 The Heisenberg Picture*

    • 11.7 Examples

      • 11.7.1 Time Development Example



    • 11.8 Sample Test Problems



  • 12 Extending QM to Two Particles and Three Dimensions

    • 12.1 Quantum Mechanics for Two Particles

    • 12.2 Quantum Mechanics in Three Dimensions

    • 12.3 Two Particles in Three Dimensions

    • 12.4 Identical Particles

    • 12.5 Sample Test Problems



  • 13 3D Problems Separable in Cartesian Coordinates

    • 13.1 Particle in a 3D Box

      • 13.1.1 Filling the Box with Fermions

      • 13.1.2 Degeneracy Pressure in Stars



    • 13.2 The 3D Harmonic Oscillator

    • 13.3 Sample Test Problems



  • 14 Angular Momentum

    • 14.1 Rotational Symmetry

    • 14.2 Angular Momentum Algebra: Raising and Lowering Operators

    • 14.3 The Angular Momentum Eigenfunctions

      • 14.3.1 Parity of the Spherical Harmonics



    • 14.4 Derivations and Computations

      • 14.4.1 Rotational Symmetry Implies Angular Momentum Conservation

      • 14.4.2 The Commutators of the Angular Momentum Operators

        • 14.4.3 Rewritingp

          • 2 μUsingL





      • 14.4.4 Spherical Coordinates and the Angular Momentum Operators.

      • 14.4.5 The OperatorsL±



    • 14.5 Examples

      • 14.5.1 The Expectation Value ofLz

      • 14.5.2 The Expectation Value ofLx



    • 14.6 Sample Test Problems



  • 15 The Radial Equation and Constant Potentials*

    • 15.1 The Radial Equation*.

    • 15.2 Behavior at the Origin*.

    • 15.3 Spherical Bessel Functions*.

    • 15.4 Particle in a Sphere*

    • 15.5 Bound States in a Spherical Potential Well*

    • 15.6 Partial Wave Analysis of Scattering*

    • 15.7 Scattering from a Spherical Well*

    • 15.8 The Radial Equation foru(r) =rR(r)*.

    • 15.9 Sample Test Problems



  • 16 Hydrogen

    • 16.1 The Radial Wavefunction Solutions

    • 16.2 The Hydrogen Spectrum

    • 16.3 Derivations and Calculations

      • 16.3.1 Solution of Hydrogen Radial Equation*.

      • 16.3.2 Computing the Radial Wavefunctions*



    • 16.4 Examples

      • 16.4.1 Expectation Values in Hydrogen States

      • 16.4.2 The Expectation of^1 rin the Ground State

      • 16.4.3 The Expectation Value ofrin the Ground State

      • 16.4.4 The Expectation Value ofvrin the Ground State



    • 16.5 Sample Test Problems



  • 17 3D Symmetric HO in Spherical Coordinates*

  • 18 Operators Matrices and Spin

    • 18.1 The Matrix Representation of Operators and Wavefunctions

    • 18.2 The Angular Momentum Matrices*.

    • 18.3 Eigenvalue Problems with Matrices

    • 18.4 Anℓ= 1 System in a Magnetic Field*

    • 18.5 Splitting the Eigenstates with Stern-Gerlach

    • 18.6 Rotation operators forℓ= 1*.

    • 18.7 A Rotated Stern-Gerlach Apparatus*

    • 18.8 Spin

    • 18.9 Other Two State Systems*

      • 18.9.1 The Ammonia Molecule (Maser)

      • 18.9.2 The Neutral Kaon System*



    • 18.10Examples

      • 18.10.1Harmonic Oscillator Hamiltonian Matrix

      • 18.10.2Harmonic Oscillator Raising Operator

      • 18.10.3Harmonic Oscillator Lowering Operator

      • 18.10.4Eigenvectors ofLx

      • 18.10.5A 90 degree rotation about the z axis.

      • 18.10.6Energy Eigenstates of anℓ= 1 System in a B-field

      • 18.10.7A series of Stern-Gerlachs

      • 18.10.8Time Development of anℓ= 1 System in a B-field: Version I

      • 18.10.9Expectation ofSxin General Spin^12 State

      • 18.10.10Eigenvectors ofSxfor Spin

      • 18.10.11Eigenvectors ofSyfor Spin

      • 18.10.12Eigenvectors ofSu

      • 18.10.13Time Development of a Spin^12 State in a B field

      • 18.10.14Nuclear Magnetic Resonance (NMR and MRI)



    • 18.11Derivations and Computations

      • 18.11.1Theℓ= 1 Angular Momentum Operators*

      • 18.11.2Compute [Lx,Ly] Using Matrices*.

      • 18.11.3Derive the Expression for Rotation OperatorRz*

      • 18.11.4Compute theℓ= 1 Rotation OperatorRz(θz)*.

      • 18.11.5Compute theℓ= 1 Rotation OperatorRy(θy)*

      • 18.11.6Derive Spin^12 Operators

      • 18.11.7Derive Spin^12 Rotation Matrices*.

      • 18.11.8NMR Transition Rate in a Oscillating B Field



    • 18.12Homework Problems

    • 18.13Sample Test Problems



  • 19 Homework Problems 130A

    • 19.1 HOMEWORK

    • 19.2 Homework

    • 19.3 Homework

    • 19.4 Homework

    • 19.5 Homework

    • 19.6 Homework

    • 19.7 Homework

    • 19.8 Homework

    • 19.9 Homework



  • 20 Electrons in an Electromagnetic Field

    • 20.1 Review of the Classical Equations of Electricity and Magnetism in CGS Units

    • 20.2 The Quantum Hamiltonian Including a B-field

    • 20.3 Gauge Symmetry in Quantum Mechanics

    • 20.4 Examples

      • 20.4.1 The Naive Zeeman Splitting

      • 20.4.2 A Plasma in a Magnetic Field



    • 20.5 Derivations and Computations

      • 20.5.1 Deriving Maxwell’s Equations for the Potentials

      • 20.5.2 The Lorentz Force from the Classical Hamiltonian

      • 20.5.3 The Hamiltonian in terms of B

      • 20.5.4 The Size of the B field Terms in Atoms

      • 20.5.5 Energy States of Electrons in a Plasma I

      • 20.5.6 Energy States of Electrons in a Plasma II

      • 20.5.7 A Hamiltonian Invariant Under Wavefunction Phase (or Gauge)Transformations

      • 20.5.8 Magnetic Flux Quantization from Gauge Symmetry



    • 20.6 Homework Problems

    • 20.7 Sample Test Problems



  • 21 Addition of Angular Momentum

    • 21.1 Adding the Spins of Two Electrons

    • 21.2 Total Angular Momentum and The Spin Orbit Interaction

    • 21.3 Adding Spin^12 to Integer Orbital Angular Momentum

    • 21.4 Spectroscopic Notation

    • 21.5 General Addition of Angular Momentum: The Clebsch-Gordan Series

    • 21.6 Interchange Symmetry for States with Identical Particles

    • 21.7 Examples

      • 21.7.1 Counting states forℓ= 3 Plus spin

      • 21.7.2 Counting states for ArbitraryℓPlus spin

      • 21.7.3 Addingℓ= 4 toℓ=

      • 21.7.4 Two electrons in an atomic P state

      • 21.7.5 The parity of the pion fromπd→nn.



    • 21.8 Derivations and Computations

      • 21.8.1 Commutators of Total Spin Operators

      • 21.8.2 Using the Lowering Operator to Find Total Spin States

      • 21.8.3 Applying theS^2 Operator toχ 1 mandχ

      • 21.8.4 Adding anyℓplus spin

      • 21.8.5 Counting the States for|ℓ 1 −ℓ 2 |≤j≤ℓ 1 +ℓ



    • 21.9 Homework Problems

    • 21.10Sample Test Problems



  • 22 Time Independent Perturbation Theory

    • 22.1 The Perturbation Series

    • 22.2 Degenerate State Perturbation Theory

    • 22.3 Examples

      • 22.3.1 H.O. with anharmonic perturbation (ax^4 ).

      • 22.3.2 Hydrogen Atom Ground State in a E-field, the Stark Effect.

      • 22.3.3 The Stark Effect for n=2 Hydrogen.



    • 22.4 Derivations and Computations

      • 22.4.1 Derivation of 1st and 2nd Order Perturbation Equations

      • 22.4.2 Derivation of 1st Order Degenerate Perturbation Equations



    • 22.5 Homework Problems

    • 22.6 Sample Test Problems



  • 23 Fine Structure in Hydrogen

    • 23.1 Hydrogen Fine Structure

    • 23.2 Hydrogen Atom in a Weak Magnetic Field

    • 23.3 Examples

    • 23.4 Derivations and Computations

      • 23.4.1 The Relativistic Correction

      • 23.4.2 The Spin-Orbit Correction

      • 23.4.3 Perturbation Calculation for Relativistic Energy Shift

      • 23.4.4 Perturbation Calculation for H2 Energy Shift

      • 23.4.5 The Darwin Term

      • 23.4.6 The Anomalous Zeeman Effect



    • 23.5 Homework Problems

    • 23.6 Sample Test Problems



  • 24 Hyperfine Structure

    • 24.1 Hyperfine Splitting

    • 24.2 Hyperfine Splitting in a B Field

    • 24.3 Examples

      • 24.3.1 Splitting of the Hydrogen Ground State

      • 24.3.2 Hyperfine Splitting in a Weak B Field

      • 24.3.3 Hydrogen in a Strong B Field

      • 24.3.4 Intermediate Field

      • 24.3.5 Positronium

      • 24.3.6 Hyperfine and Zeeman for H, muonium, positronium



    • 24.4 Derivations and Computations

      • 24.4.1 Hyperfine Correction in Hydrogen



    • 24.5 Homework Problems

    • 24.6 Sample Test Problems



  • 25 The Helium Atom

    • 25.1 General Features of Helium States

    • 25.2 The Helium Ground State

    • 25.3 The First Excited State(s)

    • 25.4 The Variational Principle (Rayleigh-Ritz Approximation)

    • 25.5 Variational Helium Ground State Energy

    • 25.6 Examples

      • 25.6.1 1D Harmonic Oscillator

      • 25.6.2 1-D H.O. with exponential wavefunction



    • 25.7 Derivations and Computations

      • 25.7.1 Calculation of the ground state energy shift



    • 25.8 Homework Problems

    • 25.9 Sample Test Problems



  • 26 Atomic Physics

    • 26.1 Atomic Shell Model

    • 26.2 The Hartree Equations

    • 26.3 Hund’s Rules

    • 26.4 The Periodic Table

    • 26.5 The Nuclear Shell Model

    • 26.6 Examples

      • 26.6.1 Boron Ground State

      • 26.6.2 Carbon Ground State

      • 26.6.3 Nitrogen Ground State

      • 26.6.4 Oxygen Ground State



    • 26.7 Homework Problems

    • 26.8 Sample Test Problems



  • 27 Molecular Physics

    • 27.1 The H+ 2 Ion

    • 27.2 The H 2 Molecule

    • 27.3 Importance of Unpaired Valence Electrons

    • 27.4 Molecular Orbitals

    • 27.5 Vibrational States

    • 27.6 Rotational States

    • 27.7 Examples

    • 27.8 Derivations and Computations

    • 27.9 Homework Problems

    • 27.10Sample Test Problems



  • 28 Time Dependent Perturbation Theory

    • 28.1 General Time Dependent Perturbations

    • 28.2 Sinusoidal Perturbations

    • 28.3 Examples

      • 28.3.1 Harmonic Oscillator in a Transient E Field



    • 28.4 Derivations and Computations

      • 28.4.1 The Delta Function of Energy Conservation



    • 28.5 Homework Problems

    • 28.6 Sample Test Problems



  • 29 Radiation in Atoms

    • 29.1 The Photon Field in the Quantum Hamiltonian

    • 29.2 Decay Rates for the Emission of Photons

    • 29.3 Phase Space: The Density of Final States

    • 29.4 Total Decay Rate Using Phase Space

    • 29.5 Electric Dipole Approximation and Selection Rules

    • 29.6 Explicit 2p to 1s Decay Rate

    • 29.7 General Unpolarized Initial State

    • 29.8 Angular Distributions

    • 29.9 Vector Operators and the Wigner Eckart Theorem

    • 29.10Exponential Decay

    • 29.11Lifetime and Line Width

      • 29.11.1Other Phenomena Influencing Line Width



    • 29.12Phenomena of Radiation Theory

      • 29.12.1The M ̈ossbauer Effect

      • 29.12.2LASERs



    • 29.13Examples

      • 29.13.1The 2P to 1S Decay Rate in Hydrogen



    • 29.14Derivations and Computations

      • 29.14.1Energy in Field for a Given Vector Potential

      • 29.14.2General Phase Space Formula

      • 29.14.3Estimate of Atomic Decay Rate



    • 29.15Homework Problems

    • 29.16Sample Test Problems



  • 30 Scattering

    • 30.1 Scattering from a Screened Coulomb Potential

    • 30.2 Scattering from a Hard Sphere

    • 30.3 Homework Problems

    • 30.4 Sample Test Problems



  • 31 Classical Scalar Fields

    • 31.1 Simple Mechanical Systems and Fields

    • 31.2 Classical Scalar Field in Four Dimensions



  • 32 Classical Maxwell Fields

    • 32.1 Rationalized Heaviside-Lorentz Units

    • 32.2 The Electromagnetic Field Tensor

    • 32.3 The Lagrangian for Electromagnetic Fields

    • 32.4 Gauge Invariance can Simplify Equations



  • 33 Quantum Theory of Radiation

    • 33.1 Transverse and Longitudinal Fields

    • 33.2 Fourier Decomposition of Radiation Oscillators

    • 33.3 The Hamiltonian for the Radiation Field

    • 33.4 Canonical Coordinates and Momenta

    • 33.5 Quantization of the Oscillators

    • 33.6 Photon States

    • 33.7 Fermion Operators

    • 33.8 Quantized Radiation Field

    • 33.9 The Time Development of Field Operators

    • 33.10Uncertainty Relations and RMS Field Fluctuations

    • 33.11Emission and Absorption of Photons by Atoms

    • 33.12Review of Radiation of Photons

      • 33.12.1Beyond the Electric Dipole Approximation



    • 33.13Black Body Radiation Spectrum



  • 34 Scattering of Photons

    • 34.1 Resonant Scattering

    • 34.2 Elastic Scattering

    • 34.3 Rayleigh Scattering

    • 34.4 Thomson Scattering

    • 34.5 Raman Effect



  • 35 Electron Self Energy Corrections

    • 35.1 The Lamb Shift



  • 36 Dirac Equation

    • 36.1 Dirac’s Motivation

    • 36.2 The Schr ̈odinger-Pauli Hamiltonian

    • 36.3 The Dirac Equation

    • 36.4 The Conserved Probability Current

    • 36.5 The Non-relativistic Limit of the Dirac Equation

      • 36.5.1 The Two Component Dirac Equation

      • 36.5.2 The Large and Small Components of the Dirac Wavefunction

      • 36.5.3 The Non-Relativistic Equation



    • 36.6 Solution of Dirac Equation for a Free Particle

      • 36.6.1 Dirac Particle at Rest

      • 36.6.2 Dirac Plane Wave Solution

      • 36.6.3 Alternate Labeling of the Plane Wave Solutions



    • 36.7 “Negative Energy” Solutions: Hole Theory

    • 36.8 Equivalence of a Two Component Theory

    • 36.9 Relativistic Covariance

    • 36.10Parity

    • 36.11Bilinear Covariants

    • 36.12Constants of the Motion for a Free Particle

    • 36.13The Relativistic Interaction Hamiltonian

    • 36.14Phenomena of Dirac States

      • 36.14.1Velocity Operator and Zitterbewegung

      • 36.14.2Expansion of a State in Plane Waves

      • 36.14.3The Expected Velocity and Zitterbewegung



    • 36.15Solution of the Dirac Equation for Hydrogen

    • 36.16Thomson Scattering

    • 36.17Hole Theory and Charge Conjugation

    • 36.18Charge Conjugate Waves

    • 36.19Quantization of the Dirac Field

    • 36.20The Quantized Dirac Field with Positron Spinors

    • 36.21Vacuum Polarization

    • 36.22The QED LaGrangian and Gauge Invariance

    • 36.23Interaction with a Scalar Field



  • 37 Formulas

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