Paradigms in Physics: Central Forces | 2024-Spring

Course Name
Paradigms in Physics: Central Forces
Course Number
PH 426 / PH 526
Year/Term
2024
Display other available years here.
Course Credits
4
Class meeting times
9 hours of lecture/discussion per week for five weeks.
Prerequisites
PH213, PH425, PH422
Course description
Gravitational and electrostatic forces; angular momentum and spherical harmonics, separation of variables in classical and quantum mechanics, hydrogen atom.

Topic/Day

Activities

Resources

Homework Due

Unit: Classical Mechanics Orbits
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Classical Mechanics Orbits
W1 D1
Introduction
Introduction to Central Forces
Central Forces Introduction Lecture Notes
OneNote Lecture Notes
Center of Mass
Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Systems of Particles
Newton's 2nd Law SWBQ
Systems of Particles Lecture Notes
Taylor Chap. 3 (Optional)
W1 D2
Angular Momentum
Angular Momentum SWBQ
Conservation of
Classical Angular Momentum
Conservation of Angular Momentum Lecture Notes
The Two-Body Problem
Taylor 8.1
Equations of Motion
Equations of Motion Lecture Notes
Taylor 8.3
Spherical/Polar Coordinates
(Review)
Curvilinear Coordinates Introduction
Curvilinear Basis Vectors
Video: Basis Vectors (Review from PH 335/422)
Velocity, Acceleration, and Angular Momentum
in Polar Coordinates
Velocity and Acceleration in Polar Coordinates
Angular Momentum in Polar Coordinates
Polar Basis Vectors
Acceleration in Polar Coordinates
Taylor 8.2
Review from PH 335:
Video: Time Derivatives of Polar Basis Vectors
Central Force Lagrangians
Reduced Mass
Reduced Mass Lecture Notes
Taylor 8.2
W1 D3
Energy
Equations of Motion from Energy Conservation Lecture Notes
Effective Potentials
Acting Out Effective Potentials
Effective Potentials
Taylor 8.4 (Optional)
GMM: Effective Potentials
W1 D4
More Effective Potentials
Effective Potentials
Change of Variables
Potentials
Air Hockey
Solutions of the Harmonic Oscillator Equation (Review)
Review from ODEs and Oscillations and Waves:}
Video: Linear ODEs w/ constant coefficients, homogeneous
Video: Linear ODEs w/ constant coefficients, inhomogeneous
Orbit Shape
Orbit Shape Lecture Notes
Taylor 8.5
W1 D5
Polar Graphs
Circles, Ellipses, Hyperbolas
Polar Plots SWBQ
Conic Sections
Visualization: Graphs of Conic Sections
W2 D1
Inverse Square Forces:
Kepler's First Law
Keplerian Orbits and Effective Potentials
GMM: Effective Potentials w/ Orbits Taylor 8.6-8.7
Orbits for Inverse Square Forces
Kepler's First Law
Taylor 8.6-8.7
W2 D2
Kepler's Second Law
Kepler's Second Law
NASA: Orbits and Kepler's Laws
Scattering
Orbits Summary
Classical Orbits Summary
Unit: Quantum Mechanics on the Ring
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Quantum Mechanics on a Ring
W2 D3
Introduction to Particle on a Ring
Intro to Particle on a Ring
Time Dependence Review
Time Evolution Refresher (Mini-Lecture)
Time Evolution of the ISW
Eigenstates on a Quantum Ring
Course Notes 16
McIntyre 7.5
W2 D4
Probabilities for the Ring from Multiple Representations
Working with Representations on the Ring
Course Notes 17
Video: Matrix Representations for a Particle on a Ring
Video: Position Representations for a Particle on a Ring
Video: Degeneracy on the Ring
Meaning of the Wave Function
Wavefunctions on a Quantum Ring
Histograms for the Quantum Ring
Time Dependence for a Quantum Particle on a Ring Part 2
Normalization of Eigenstates on the Ring
Course Notes 16
W2 D5
Exponential Form of Fourier Series
GMM: Alternative Forms of Fourier Series
Video: Exponential Form of Fourier Series, aka Quantum States on a Ring
Time Dependence on a Ring
Time Dependence for a Quantum Particle on a Ring Part 1
Visualization of Quantum Probabilities for a Particle Confined to a Ring
Course Notes 18
GMM: State on a Ring Geogebra Applet
Extra Time Dependent Ring Activities
Expectation Values for a Particle on a Ring
Time Dependence for a Particle on a Ring
Expectation Values for a Particle on a Ring
Unit: Math Bits (Power Series Solutions of ODEs)
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Power Series Solutions of ODEs (Math Bits)
W3 D1
To prepare for class:
Review Power Series
Power Series Review
Using Technology to Explore Power Series Approximations
Sensemaking from Graphs of Power Series I
Visualization of Power Series Approximations
(Review Power Series)
GMM: Power Series
Math Bits:
Series Solutions
Power Series Solution of ODEs: Math Example
GMM: Power Series Solutions: Method/Example
W3 D2
Math Bits: Series Solutions for Legendre Polynomials
Series Solutions of Legendre's Equation
Course Notes 21
McIntyre 7.6.1
Math Bits: Theorems about Power Series Solutions
GMM: Power Series Solutions: Theorems
Math Bits: Power Series Solutions
around Singular Points
(Frobenius Series)
Math Bits: Properties of Legendre Polynomials
Legendre Polynomials Lecture
Course Notes 22
Wikipedia article on Legendre Polynomials
Math Bits: Legendre's Associated Equation
McIntyre 7.6.2
Unit: Math Bits (PDEs)
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Partial Differential Equations (Math Bits)
W3 D3
Math Bits: Vector Spaces (Review)
Before Class:
Review Vector Spaces
GMM: Definition of a Vector Space
Video: Vector Spaces
Math Bits: Inner Products (Review)
Before Class:
Review Inner Products
GMM: Definition and Properties of an Inner Product
Video: Inner Products
GMM: Visualizing the Dot Product in Higher Dimensions
GMM: Visualizing the Product of Two Functions
GMM: Inner Products of Functions
W3 D4
Math Bits: Separation of Variables
Rectangular
(Review)
Math Bits: Separating out Center-of-Mass: Quantum
Course Notes 12-13
McIntyre 7.1
Math Bits Example:
Particle in a 2-D Quantum Box
After Class:
QuVis Simulation: 2-D Rectangular Box
QuVis Simulation: 2-D Circular Box
Math Bits Example:
Quantum Cylinder
Quantum Cylinder
Math Bits:
Separation of Variables in Curvilinear Coordinates
Course Notes 14, 15, 19
McIntyre 7.4
GMM: Separation of Variables
W3 D5
Math Bits:
PDEs in Physics
GMM: Important PDEs in Physics
GMM: Classification of PDEs
Video: Classification of PDEs
Math Bits:
Boundary & Initial Conditions
GMM: Boundary & Initial Conditions on PDEs in Physics
Math Bits:
Sturm-Liouville Theory
Sturm-Liouville Theorems
GMM: Sturm-Liouville Theory
Video: Introduction to Sturm-Liouville Theory
Math Bits:
Legendre Polynomial Series
Legendre Polynomial Series Lecture
Guessing Legendre Coefficients
Expanding a Function in Legendre Polynomials.
Course Notes 23-24
GMM: Legendre Polynomial Series
W4 D1
MEMORIAL DAY
No Class
Unit: Angular Momentum in Quantum Mechanics
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Angular Momentum in Quantum Mechanics
W4 D2
Quantum Ring with Separation of Variables
Time Evolution of a Quantum Particle on a Ring with Arms
McIntyre 7.5
Schrödinger's Equation:
Spherical Symmetry
Course Notes 2, Sections 15-16
McIntyre 7.2
Change of Variables (Review)
Course Notes 20
Video: Change of Variables in Legendre's Equation
Rigid Rotor
Course Notes 25
McIntyre 7.4
McIntyre 7.6
Video: Separation of Variables for the Rigid Rotor
W4 D3
Spherical Harmonics:
Properties
Spherical Harmonics
McIntyre 7.6.4
Spherical Harmonics:
Visualization
Visualizing Spherical Harmonics
Visualizing Combinations of Spherical Harmonics
Spherical Harmonics on a Balloon
Video: Description of Spherical Harmonics
McIntyre 7.6.5
W4 D4
Quantum States of the Rigid Rotor
Spherical Harmonics on a Balloon
Quantum Systems Reference Sheet
Probabilities for a Quantum Particle on a Unit Sphere in Spherical Harmonic Functions
Finding Coefficients of a Spherical Harmonics Series
W4 D5
Rigid Rotor Probabilities & Degeneracy
Probabilities for a Quantum Particle on a Unit Sphere in Spherical Harmonic Functions
W5 D1
Symmetries, Parity, Degeneracy
Time Dependence
Quantum Particle on a Unit Sphere in Dirac Notation
McIntyre 7.6.3
Orbital Angular Momentum:
Spin 1 revisited
Matrix Representation of Angular Momentum
McIntyre 7.3
Raising and Lowering Operators
Raising and Lowering Operators for Spin
Griffiths 4.3
Unit: Quantum Mechanics of the Hydrogen Atom
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Quantum Mechanics of the (Unperturbed) Hydrogen Atom
W5 D2
Commutation Relations
Angular Momentum Commutation Relations: Lecture
The Radial Equation
McIntyre 8.1-8.4
Krane: Class Notes 3
W5 D3
Asymptotic Behavior
W5 D4
Radial Functions
Radial Functions for a Hydrogenic Atom
Visualizing Radial Functions
Visualizing Radial Wave Functions
Video: Visualization of Radial Functions
Laguerre Polynomals
Hydrogen Atom Wave Functions
McIntyre 8.5-8.6
Calculations on Hydrogen Atom States
Quantum Calculations on the Hydrogen Atom
Hydrogen Probabilities in Matrix Notation
Hydrogen Spectra
McIntyre 8.3
Symmetries, Parity, Degeneracy
Hydrogen Atom Visualization of Probability Densities
Visualization of Quantum Probabilities for the Hydrogen Atom
Video: Interpreting Hydrogen Atom Plots
Classical Limit
Applications
Probability of Finding an Electron Inside the Bohr Radius
W5 D5
Review
Classical Orbits Equation Sheet
Quantum Ring Equation Sheet
Quantum Systems Reference Sheet
Spherical Harmonics
Radial Functions for a Hydrogenic Atom
McIntyre Appendix F Integrals
Final Exam
FINAL EXAM