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Contents
Contents
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
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
The Solar Temperature
*
Black Body Radiation from the Early Universe
*
Compton Scattering
*
Rutherford's Nuclear Size
*
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
Fourier Series
*
Fourier Transform
*
Integral of Gaussian
Fourier Transform of Gaussian
*
Time Dependence of a Gaussian Wave Packet
*
Numbers
The Dirac Delta Function
Examples
The Square Wave Packet
The Gaussian Wave Packet
*
The Dirac Delta Function Wave Packet
*
Can I ``See'' inside an Atom
Can I ``See'' inside a Nucleus
Estimate the Hydrogen Ground State Energy
Sample Test Problems
Operators
Operators in Position Space
The Momentum Operator
The Energy Operator
The Position Operator
The Hamiltonian Operator
Operators in Momentum Space
Expectation Values
Dirac Bra-ket Notation
Commutators
Derivations and Computations
Verify Momentum Operator
Verify Energy Operator
Examples
Expectation Value of Momentum in a Given State
Commutator of
and
Commutator of
and
Commutator of
and
Commutator of
and
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
Sample Test Problems
One Dimensional Potentials
Piecewise Constant Potentials in 1D
The General Solution for a Constant Potential
The Potential Step
The Potential Well with
*
Bound States in a Potential Well
*
The Potential Barrier
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
Probability Flux for the Potential Step
*
Scattering from a 1D Potential Well
*
Bound States of a 1D Potential Well
*
Solving the HO Differential Equation
*
1D Model of a Molecule Derivation
*
1D Model of a Crystal Derivation
*
Examples
Sample Test Problems
Harmonic Oscillator Solution using Operators
Introducing
and
Commutators of
,
and
Use Commutators to Derive HO Energies
Raising and Lowering Constants
Expectation Values of
and
The Wavefunction for the HO Ground State
Examples
The expectation value of
in eigenstate
The expectation value of
in eigenstate
The expectation value of
in the state
.
The expectation value of
in eigenstate
The expectation value of
in eigenstate
Time Development Example
Sample Test Problems
More Fun with Operators
Operators in a Vector Space
Review of Operators
Projection Operators
and Completeness
Unitary Operators
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
Rotational Symmetry Implies Angular Momentum Conservation
The Commutators of the Angular Momentum Operators
Rewriting
Using
Spherical Coordinates and the Angular Momentum Operators
The Operators
Examples
The Expectation Value of
The Expectation Value of
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
*
Sample Test Problems
Hydrogen
The Radial Wavefunction Solutions
The Hydrogen Spectrum
Derivations and Calculations
Solution of Hydrogen Radial Equation
*
Computing the Radial Wavefunctions
*
Examples
Expectation Values in Hydrogen States
The Expectation of
in the Ground State
The Expectation Value of
in the Ground State
The Expectation Value of
in the Ground State
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
System in a Magnetic Field
*
Splitting the Eigenstates with Stern-Gerlach
Rotation operators for
*
A Rotated Stern-Gerlach Apparatus
*
Spin
Other Two State Systems
*
The Ammonia Molecule (Maser)
The Neutral Kaon System
*
Examples
Harmonic Oscillator Hamiltonian Matrix
Harmonic Oscillator Raising Operator
Harmonic Oscillator Lowering Operator
Eigenvectors of
A 90 degree rotation about the z axis.
Energy Eigenstates of an
System in a B-field
A series of Stern-Gerlachs
Time Development of an
System in a B-field: Version I
Expectation of
in General Spin
State
Eigenvectors of
for Spin
Eigenvectors of
for Spin
Eigenvectors of
Time Development of a Spin
State in a B field
Nuclear Magnetic Resonance (NMR and MRI)
Derivations and Computations
The
Angular Momentum Operators
*
Compute
Using Matrices
*
Derive the Expression for Rotation Operator
*
Compute the
Rotation Operator
*
Compute the
Rotation Operator
*
Derive Spin
Operators
Derive Spin
Rotation Matrices
*
NMR Transition Rate in a Oscillating B Field
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
Deriving Maxwell's Equations for the Potentials
The Lorentz Force from the Classical Hamiltonian
The Hamiltonian in terms of B
The Size of the B field Terms in Atoms
Energy States of Electrons in a Plasma I
Energy States of Electrons in a Plasma II
A Hamiltonian Invariant Under Wavefunction Phase (or Gauge) Transformations
Magnetic Flux Quantization from Gauge Symmetry
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
to Integer Orbital Angular Momentum
Spectroscopic Notation
General Addition of Angular Momentum: The Clebsch-Gordan Series
Interchange Symmetry for States with Identical Particles
Examples
Counting states for
Plus spin
Counting states for Arbitrary
Plus spin
Adding
to
Two electrons in an atomic P state
The parity of the pion from
.
Derivations and Computations
Commutators of Total Spin Operators
Using the Lowering Operator to Find Total Spin States
Applying the
Operator to
and
.
Adding any
plus spin
.
Counting the States for
.
Homework Problems
Sample Test Problems
Time Independent Perturbation Theory
The Perturbation Series
Degenerate State Perturbation Theory
Examples
H.O. with anharmonic perturbation (
).
Hydrogen Atom Ground State in a E-field, the Stark Effect.
The Stark Effect for n=2 Hydrogen.
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
The Relativistic Correction
The Spin-Orbit Correction
Perturbation Calculation for Relativistic Energy Shift
Perturbation Calculation for H2 Energy Shift
The Darwin Term
The Anomalous Zeeman Effect
Homework Problems
Sample Test Problems
Hyperfine Structure
Hyperfine Splitting
Hyperfine Splitting in a B Field
Examples
Splitting of the Hydrogen Ground State
Hyperfine Splitting in a Weak B Field
Hydrogen in a Strong B Field
Intermediate Field
Positronium
Hyperfine and Zeeman for H, muonium, positronium
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
Boron Ground State
Carbon Ground State
Nitrogen Ground State
Oxygen Ground State
Homework Problems
Sample Test Problems
Molecular Physics
The
Ion
The
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
Energy in Field for a Given Vector Potential
General Phase Space Formula
Estimate of Atomic Decay Rate
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
Classical Maxwell Fields
Rationalized Heaviside-Lorentz Units
The Electromagnetic Field Tensor
The Lagrangian for Electromagnetic Fields
Gauge Invariance can Simplify Equations
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
Scattering of Photons
Resonant Scattering
Elastic Scattering
Rayleigh Scattering
Thomson Scattering
Raman Effect
Electron Self Energy Corrections
The Lamb Shift
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
Formulas
About this document ...
Jim Branson 2013-04-22