Quantum Physics 130

Quantum Physics

(UCSD Physics 130)

April 2, 2003

Contents
Preface
Course Summary
- Problems with Classical Physics
- Thought Experiments on Diffraction
- Probability Amplitudes
- Wave Packets and Uncertainty
- Operators
- Expectation Values
- Commutators
- The Schrödinger Equation
- Eigenfunctions, Eigenvalues and Vector Spaces
- A Particle in a Box
- Piecewise Constant Potentials in One Dimension
- The Harmonic Oscillator in One Dimension
- Delta Function Potentials in One Dimension
- Harmonic Oscillator Solution with Operators
- More Fun with Operators
- Two Particles in 3 Dimensions
- Identical Particles
- Some 3D Problems Separable in Cartesian Coordinates
- Angular Momentum
- Solutions to the Radial Equation for Constant Potentials
- Hydrogen
- Solution of the 3D HO Problem in Spherical Coordinates
- Matrix Representation of Operators and States
- A Study of $\bgroup\color{black}$\ell=1$\egroup$ Operators and Eigenfunctions
- Spin 1/2 and other 2 State Systems
- Quantum Mechanics in an Electromagnetic Field
- Local Phase Symmetry in Quantum Mechanics and the Gauge Symmetry
- Addition of Angular Momentum
- Time Independent Perturbation Theory
- The Fine Structure of Hydrogen
- Hyperfine Structure
- The Helium Atom
- Atomic Physics
- Molecules
- Time Dependent Perturbation Theory
- Radiation in Atoms
- Classical Field Theory
- The Classical Electromagnetic Field
- Quantization of the EM Field
- Scattering of Photons
- Electron Self Energy
- The Dirac Equation
- The Dirac Equation
The Problems with Classical Physics
- Black Body Radiation *
- The Photoelectric Effect
- The Rutherford Atom *
- Atomic Spectra *
  - The Bohr Atom *
- Derivations and Computations
  - Black Body Radiation Formulas *
  - The Fine Structure Constant and the Coulomb Potential
- Examples
- Sample Test Problems
Diffraction
- Diffraction from Two Slits
- Single Slit Diffraction
- Diffraction from Crystals
- The DeBroglie Wavelength
  - Computing DeBroglie Wavelengths
- Wave Particle Duality (Thought Experiments)
- Examples
  - Intensity Distribution for Two Slit Diffraction *
  - Intensity Distribution for Single Slit Diffraction *
- Sample Test Problems
The Solution: Probability Amplitudes
- Derivations and Computations
  - Review of Complex Numbers
  - Review of Traveling Waves
- Sample Test Problems
Wave Packets
- Building a Localized Single-Particle Wave Packet
- Two Examples of Localized Wave Packets
- The Heisenberg Uncertainty Principle
- Position Space and Momentum Space
- Time Development of a Gaussian Wave Packet *
- Derivations and Computations
- Examples
- Sample Test Problems
Operators
- Operators in Position Space
- Operators in Momentum Space
- Expectation Values
- Dirac Bra-ket Notation
- Commutators
- Derivations and Computations
  - Verify Momentum Operator
  - Verify Energy Operator
- Examples
- Sample Test Problems
The Schrödinger Equation
- Deriving the Equation from Operators
- The Flux of Probability *
- The Schrödinger Wave Equation
- The Time Independent Schrödinger Equation
- Derivations and Computations
  - Linear Operators
  - Probability Conservation Equation *
- Examples
  - Solution to the Schrödinger Equation in a Constant Potential
- Sample Test Problems
Eigenfunctions, Eigenvalues and Vector Spaces
- Eigenvalue Equations
- Hermitian Conjugate of an Operator
- Hermitian Operators
- Eigenfunctions and Vector Space
- The Particle in a 1D Box
  - The Same Problem with Parity Symmetry
- Momentum Eigenfunctions
- Derivations and Computations
  - Eigenfunctions of Hermitian Operators are Orthogonal
  - Continuity of Wavefunctions and Derivatives
- Examples
  - Hermitian Conjugate of a Constant Operator
  - Hermitian Conjugate of $\bgroup\color{black}${\partial\over\partial x}$\egroup$
- Sample Test Problems
One Dimensional Potentials
- Piecewise Constant Potentials in 1D
- The 1D Harmonic Oscillator
- The Delta Function Potential *
- The Delta Function Model of a Molecule *
- The Delta Function Model of a Crystal *
- The Quantum Rotor
- Derivations and Computations
- Examples
- Sample Test Problems
Harmonic Oscillator Solution using Operators
- Introducing $\bgroup\color{black}$A$\egroup$ and $\bgroup\color{black}$A^\dagger$\egroup$
- Commutators of $\bgroup\color{black}$A$\egroup$ , $\bgroup\color{black}$A^\dagger$\egroup$ and $\bgroup\color{black}$H$\egroup$
- Use Commutators to Derive HO Energies
  - Raising and Lowering Constants
- Expectation Values of $\bgroup\color{black}$p$\egroup$ and $\bgroup\color{black}$x$\egroup$
- The Wavefunction for the HO Ground State
- Examples
- Sample Test Problems
More Fun with Operators
- Operators in a Vector Space
- A Complete Set of Mutually Commuting Operators
- Uncertainty Principle for Non-Commuting Operators
- Time Derivative of Expectation Values *
- The Time Development Operator *
- The Heisenberg Picture *
- Examples
  - Time Development Example
- Sample Test Problems
Extending QM to Two Particles and Three Dimensions
- Quantum Mechanics for Two Particles
- Quantum Mechanics in Three Dimensions
- Two Particles in Three Dimensions
- Identical Particles
- Sample Test Problems
3D Problems Separable in Cartesian Coordinates
- Particle in a 3D Box
  - Filling the Box with Fermions
  - Degeneracy Pressure in Stars
- The 3D Harmonic Oscillator
- Sample Test Problems
Angular Momentum
- Rotational Symmetry
- Angular Momentum Algebra: Raising and Lowering Operators
- The Angular Momentum Eigenfunctions
  - Parity of the Spherical Harmonics
- Derivations and Computations
- Examples
  - The Expectation Value of $\bgroup\color{black}$L_z$\egroup$
  - The Expectation Value of $\bgroup\color{black}$L_x$\egroup$
- Sample Test Problems
The Radial Equation and Constant Potentials *
- The Radial Equation *
- Behavior at the Origin *
- Spherical Bessel Functions *
- Particle in a Sphere *
- Bound States in a Spherical Potential Well *
- Partial Wave Analysis of Scattering *
- Scattering from a Spherical Well *
- The Radial Equation for $\bgroup\color{black}$u(r)=rR(r)$\egroup$ *
- Sample Test Problems
Hydrogen
- The Radial Wavefunction Solutions
- The Hydrogen Spectrum
- Derivations and Calculations
  - Solution of Hydrogen Radial Equation *
  - Computing the Radial Wavefunctions *
- Examples
- Sample Test Problems
3D Symmetric HO in Spherical Coordinates *
Operators Matrices and Spin
- The Matrix Representation of Operators and Wavefunctions
- The Angular Momentum Matrices*
- Eigenvalue Problems with Matrices
- An $\bgroup\color{black}$\ell=1$\egroup$ System in a Magnetic Field*
- Splitting the Eigenstates with Stern-Gerlach
- Rotation operators for $\bgroup\color{black}$\ell=1$\egroup$ *
- A Rotated Stern-Gerlach Apparatus*
- Spin $\bgroup\color{black}${1\over 2}$\egroup$
- Other Two State Systems*
  - The Ammonia Molecule (Maser)
  - The Neutral Kaon System*
- Examples
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Homework Problems 130A
- HOMEWORK 1
- Homework 2
- Homework 3
- Homework 4
- Homework 5
- Homework 6
- Homework 7
- Homework 8
- Homework 9
Electrons in an Electromagnetic Field
- Review of the Classical Equations of Electricity and Magnetism in CGS Units
- The Quantum Hamiltonian Including a B-field
- Gauge Symmetry in Quantum Mechanics
- Examples
  - The Naive Zeeman Splitting
  - A Plasma in a Magnetic Field
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Addition of Angular Momentum
- Adding the Spins of Two Electrons
- Total Angular Momentum and The Spin Orbit Interaction
- Adding Spin $\bgroup\color{black}${1\over 2}$\egroup$ to Integer Orbital Angular Momentum
- Spectroscopic Notation
- General Addition of Angular Momentum: The Clebsch-Gordan Series
- Interchange Symmetry for States with Identical Particles
- Examples
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Time Independent Perturbation Theory
- The Perturbation Series
- Degenerate State Perturbation Theory
- Examples
- Derivations and Computations
  - Derivation of 1st and 2nd Order Perturbation Equations
  - Derivation of 1st Order Degenerate Perturbation Equations
- Homework Problems
- Sample Test Problems
Fine Structure in Hydrogen
- Hydrogen Fine Structure
- Hydrogen Atom in a Weak Magnetic Field
- Examples
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Hyperfine Structure
- Hyperfine Splitting
- Hyperfine Splitting in a B Field
- Examples
- Derivations and Computations
  - Hyperfine Correction in Hydrogen
- Homework Problems
- Sample Test Problems
The Helium Atom
- General Features of Helium States
- The Helium Ground State
- The First Excited State(s)
- The Variational Principle (Rayleigh-Ritz Approximation)
- Variational Helium Ground State Energy
- Examples
  - 1D Harmonic Oscillator
  - 1-D H.O. with exponential wavefunction
- Derivations and Computations
  - Calculation of the ground state energy shift
- Homework Problems
- Sample Test Problems
Atomic Physics
- Atomic Shell Model
- The Hartree Equations
- Hund's Rules
- The Periodic Table
- The Nuclear Shell Model
- Examples
- Homework Problems
- Sample Test Problems
Molecular Physics
- The $\bgroup\color{black}$\mathrm{H}_2^+$\egroup$ Ion
- The $\bgroup\color{black}$\mathrm{H}_2$\egroup$ Molecule
- Importance of Unpaired Valence Electrons
- Molecular Orbitals
- Vibrational States
- Rotational States
- Examples
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Time Dependent Perturbation Theory
- General Time Dependent Perturbations
- Sinusoidal Perturbations
- Examples
  - Harmonic Oscillator in a Transient E Field
- Derivations and Computations
  - The Delta Function of Energy Conservation
- Homework Problems
- Sample Test Problems
Radiation in Atoms
- The Photon Field in the Quantum Hamiltonian
- Decay Rates for the Emission of Photons
- Phase Space: The Density of Final States
- Total Decay Rate Using Phase Space
- Electric Dipole Approximation and Selection Rules
- Explicit 2p to 1s Decay Rate
- General Unpolarized Initial State
- Angular Distributions
- Vector Operators and the Wigner Eckart Theorem
- Exponential Decay
- Lifetime and Line Width
  - Other Phenomena Influencing Line Width
- Phenomena of Radiation Theory
  - The Mössbauer Effect
  - LASERs
- Examples
  - The 2P to 1S Decay Rate in Hydrogen
- Derivations and Computations
- Homework Problems
- Sample Test Problems
Scattering
- Scattering from a Screened Coulomb Potential
- Scattering from a Hard Sphere
- Homework Problems
- Sample Test Problems
Classical Scalar Fields
- Simple Mechanical Systems and Fields
- Classical Scalar Field in Four Dimensions
- Homework Problems
Classical Maxwell Fields
- Rationalized Heaviside-Lorentz Units
- The Electromagnetic Field Tensor
- The Lagrangian for Electromagnetic Fields
- Gauge Invariance can Simplify Equations
- Homework Problems
Quantum Theory of Radiation
- Transverse and Longitudinal Fields
- Fourier Decomposition of Radiation Oscillators
- The Hamiltonian for the Radiation Field
- Canonical Coordinates and Momenta
- Quantization of the Oscillators
- Photon States
- Fermion Operators
- Quantized Radiation Field
- The Time Development of Field Operators
- Uncertainty Relations and RMS Field Fluctuations
- Emission and Absorption of Photons by Atoms
- Review of Radiation of Photons
  - Beyond the Electric Dipole Approximation
- Black Body Radiation Spectrum
- Homework Problems
Scattering of Photons
- Resonant Scattering
- Elastic Scattering
- Rayleigh Scattering
- Thomson Scattering
- Raman Effect
- Homework Problems
Electron Self Energy Corrections
- The Lamb Shift
- Homework Problems
Dirac Equation
- Dirac's Motivation
- The Schrödinger-Pauli Hamiltonian
- The Dirac Equation
- The Conserved Probability Current
- The Non-relativistic Limit of the Dirac Equation
  - The Two Component Dirac Equation
  - The Large and Small Components of the Dirac Wavefunction
  - The Non-Relativistic Equation
- Solution of Dirac Equation for a Free Particle
  - Dirac Particle at Rest
  - Dirac Plane Wave Solution
  - Alternate Labeling of the Plane Wave Solutions
- ``Negative Energy'' Solutions: Hole Theory
- Equivalence of a Two Component Theory
- Relativistic Covariance
- Parity
- Bilinear Covariants
- Constants of the Motion for a Free Particle
- The Relativistic Interaction Hamiltonian
- Phenomena of Dirac States
  - Velocity Operator and Zitterbewegung
  - Expansion of a State in Plane Waves
  - The Expected Velocity and Zitterbewegung
- Solution of the Dirac Equation for Hydrogen
- Thomson Scattering
- Hole Theory and Charge Conjugation
- Charge Conjugate Waves
- Quantization of the Dirac Field
- The Quantized Dirac Field with Positron Spinors
- Vacuum Polarization
- The QED LaGrangian and Gauge Invariance
- Interaction with a Scalar Field
- Homework Problems
Formulas
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Branson 2008-12-22