Quantum Physics (UCSD Physics 130), Exercises of Quantum Mechanics

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Quantum Physics

(UCSD Physics 130)

April 2, 2003

Contents

14.4.3 Rewriting p

• 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 of ℓ = 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 of E and t
    • 6.7.3 Commutator of E and x
    • 6.7.4 Commutator of p and xn
    • 6.7.5 Commutator of Lx and Ly
  • 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 with E > 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 Introducing A and A†
  • 10.2 Commutators of A, A† and H
  • 10.3 Use Commutators to Derive HO Energies
    • 10.3.1 Raising and Lowering Constants
  • 10.4 Expectation Values of p and x
  • 10.5 The Wavefunction for the HO Ground State
  • 10.6 Examples
    • 10.6.1 The expectation value of x in eigenstate
    • 10.6.2 The expectation value of p in eigenstate
    • 10.6.3 The expectation value of x in 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 2 pm^2 in 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
      • 2 μ Using L
    • 14.4.4 Spherical Coordinates and the Angular Momentum Operators
    • 14.4.5 The Operators L±
  • 14.5 Examples
    • 14.5.1 The Expectation Value of Lz
    • 14.5.2 The Expectation Value of Lx
  • 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 for u(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 r in the Ground State
    • 16.4.3 The Expectation Value of r in the Ground State
    • 16.4.4 The Expectation Value of vr in 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 An ℓ = 1 System in a Magnetic Field*
  • 18.5 Splitting the Eigenstates with Stern-Gerlach
  • 18.6 Rotation operators for ℓ = 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
    • 18.10.1 Harmonic Oscillator Hamiltonian Matrix
    • 18.10.2 Harmonic Oscillator Raising Operator
    • 18.10.3 Harmonic Oscillator Lowering Operator
    • 18.10.4 Eigenvectors of Lx
    • 18.10.5 A 90 degree rotation about the z axis.
    • 18.10.6 Energy Eigenstates of an ℓ = 1 System in a B-field
    • 18.10.7 A series of Stern-Gerlachs
    • 18.10.8 Time Development of an ℓ = 1 System in a B-field: Version I
    • 18.10.9 Expectation of Sx in General Spin 12 State
    • 18.10.10Eigenvectors of Sx for Spin
    • 18.10.11Eigenvectors of Sy for Spin
    • 18.10.12Eigenvectors of Su
    • 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
    • 18.11.1 The ℓ = 1 Angular Momentum Operators*
    • 18.11.2 Compute [Lx, Ly ] Using Matrices *
    • 18.11.3 Derive the Expression for Rotation Operator Rz *
    • 18.11.4 Compute the ℓ = 1 Rotation Operator Rz (θz ) *
    • 18.11.5 Compute the ℓ = 1 Rotation Operator Ry (θy ) *
    • 18.11.6 Derive Spin 12 Operators
    • 18.11.7 Derive Spin 12 Rotation Matrices *
    • 18.11.8 NMR 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 for ℓ = 3 Plus spin
    • 21.7.2 Counting states for Arbitrary ℓ Plus spin
    • 21.7.3 Adding ℓ = 4 to ℓ =
    • 21.7.4 Two electrons in an atomic P state
    • 21.7.5 The parity of the pion from πd → 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 the S^2 Operator to χ 1 m and χ
    • 21.8.4 Adding any ℓ plus 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
    • 29.11.1 Other Phenomena Influencing Line Width
  • 29.12Phenomena of Radiation Theory
    • 29.12.1 The M¨ossbauer Effect
    • 29.12.2 LASERs
  • 29.13Examples
    • 29.13.1 The 2P to 1S Decay Rate in Hydrogen
  • 29.14Derivations and Computations
    • 29.14.1 Energy in Field for a Given Vector Potential
    • 29.14.2 General Phase Space Formula
    • 29.14.3 Estimate 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
    • 33.12.1 Beyond 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
    • 36.14.1 Velocity Operator and Zitterbewegung
    • 36.14.2 Expansion of a State in Plane Waves
    • 36.14.3 The 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

has the ability to produce high quality equations and input is fast compared to other options. The LaTeX2html translator functions well enough for the conversion.

Projecting the notes can be very useful in lecture for introductions, for review, and for quick looks at derivations. The primary teaching though probably still works best at the blackboard. One thing that our classrooms really don’t facilitate is switching from one mode to the other.

In a future class, with the notes fully prepared, I will plan to decrease the formal lecture time and add lab or discussion session time, with students working moving at their own pace using computers. Projects could be worked on in groups or individually. Instructors would be available to answer questions and give suggestions.

Similar sessions would be possible at a distance. The formal lecture could be taped and available in bite size pieces inside the lecture notes. Advanced classes with small numbers of students could be taught based on notes, with less instructor support than is usual. Classes could be offered more often than is currently feasible.

Jim Branson

1 Course Summary

1.1 Problems with Classical Physics

Around the beginning of the 20th century, classical physics, based on Newtonian Mechanics and Maxwell’s equations of Electricity and Magnetism described nature as we knew it. Statistical Me- chanics was also a well developed discipline describing systems with a large number of degrees of freedom. Around that time, Einstein introduced Special Relativity which was compatible with Maxwell’s equations but changed our understanding of space-time and modified Mechanics.

Many things remained unexplained. While the electron as a constituent of atoms had been found, atomic structure was rich and quite mysterious. There were problems with classical physics, (See section 2) including Black Body Radiation, the Photoelectric effect, basic Atomic Theory, Compton Scattering, and eventually with the diffraction of all kinds of particles. Plank hypothesized that EM energy was always emitted in quanta E = hν = ¯hω

to solve the Black Body problem. Much later, deBroglie derived the wavelength (See section 3.4) for particles.

λ = h p

Ultimately, the problems led to the development of Quantum Mechanics in which all particles are understood to have both wave and a particle behavior.

1.2 Thought Experiments on Diffraction

Diffraction (See section 3) of photons, electrons, and neutrons has been observed (see the pictures) and used to study crystal structure.

To understand the experimental input in a simplified way, we consider some thought experiments on the diffraction (See section 3.5) of photons, electrons, and bullets through two slits. For example, photons, which make up all electromagnetic waves, show a diffraction pattern exactly as predicted by the theory of EM waves, but we always detect an integer number of photons with the Plank’s relation, E = hν, between wave frequency and particle energy satisfied.

Electrons, neutrons, and everything else behave in exactly the same way, exhibiting wave-like diffrac- tion yet detection of an integer number of particles and satisfying λ = h p. This deBroglie wavelength formula relates the wave property λ to the particle property p.

1.3 Probability Amplitudes

In Quantum Mechanics, we understand this wave-particle duality using (complex) probability amplitudes (See section 4) which satisfy a wave equation.

ψ(~x, t) = ei(~k·~x−ωt)^ = ei(~p·~x−Et)/¯h

It shows that due to the wave nature of particles, we cannot localize a particle into a small volume without increasing its energy. For example, we can estimate the ground state energy (and the size of) a Hydrogen atom very well from the uncertainty principle.

The next step in building up Quantum Mechanics is to determine how a wave function develops with time – particularly useful if a potential is applied. The differential equation which wave functions must satisfy is called the Schr¨odinger Equation.

1.5 Operators

The Schr¨odinger equation comes directly out of our understanding of wave packets. To get from wave packets to a differential equation, we use the new concept of (linear) operators (See section 6). We determine the momentum and energy operators by requiring that, when an operator for some variable v acts on our simple wavefunction, we get v times the same wave function.

p( xop )= ¯h i ∂x∂

p( xop )ei(~p·~x−Et)/¯h^ = ¯h i ∂x ∂ei(~p·~x−Et)/h¯^ = pxei(p~·~x−Et)/¯h

E(op)^ = i¯h ∂t∂

E(op)ei(~p·~x−Et)/¯h^ = i¯h ∂t ∂ei(~p·~x−Et)/¯h^ = Eei(~p·~x−Et)/¯h

1.6 Expectation Values

We can use operators to help us compute the expectation value (See section 6.3) of a physical variable. If a particle is in the state ψ(x), the normal way to compute the expectation value of f (x) is

〈f (x)〉 =

∫^ ∞

−∞

P (x)f (x)dx =

∫^ ∞

−∞

ψ∗(x)ψ(x)f (x)dx.

If the variable we wish to compute the expectation value of (like p) is not a simple function of x, let its operator act on ψ(x)

〈p〉 =

∫^ ∞

−∞

ψ∗(x)p(op)ψ(x)dx.

We have a shorthand notation for the expectation value of a variable v in the state ψ which is quite useful.

〈ψ|v|ψ〉 ≡

∫^ ∞

−∞

ψ∗(x)v(op)^ ψ(x)dx.

We extend the notation from just expectation values to

〈ψ|v|φ〉 ≡

∫^ ∞

−∞

ψ∗(x)v(op)φ(x)dx

and

〈ψ|φ〉 ≡

∫^ ∞

−∞

ψ∗(x)φ(x)dx

We use this shorthand Dirac Bra-Ket notation a great deal.

1.7 Commutators

Operators (or variables in quantum mechanics) do not necessarily commute. We can compute the commutator (See section 6.5) of two variables, for example

[p, x] ≡ px − xp = ¯h i.

Later we will learn to derive the uncertainty relation for two variables from their commutator. We will also use commutators to solve several important problems.

1.8 The Schr¨odinger Equation

Wave functions must satisfy the Schr¨odinger Equation (See section 7) which is actually a wave equation. −¯h^2 2 m ∇

(^2) ψ(~x, t) + V (~x)ψ(~x, t) = i¯h ∂ψ(~x, t) ∂t We will use it to solve many problems in this course. In terms of operators, this can be written as

Hψ(~x, t) = Eψ(~x, t)

where (dropping the (op) label) H = 2 pm^2 + V (~x) is the Hamiltonian operator. So the Schr¨odinger Equation is, in some sense, simply the statement (in operators) that the kinetic energy plus the potential energy equals the total energy.

1.9 Eigenfunctions, Eigenvalues and Vector Spaces

For any given physical problem, the Schr¨odinger equation solutions which separate (See section 7.4) (between time and space), ψ(x, t) = u(x)T (t), are an extremely important set. If we assume the equation separates, we get the two equations (in one dimension for simplicity)

i¯h ∂T ∂t^ (t )= E T (t)

Hu(x) = E u(x)

The second equation is called the time independent Schr¨odinger equation. For bound states, there are only solutions to that equation for some quantized set of energies

Hui(x) = Eiui(x).

For states which are not bound, a continuous range of energies is allowed.