Paradigms in Physics: Central Forces | 2026-Spring

Course Name
Paradigms in Physics: Central Forces
Course Number
PH 426 / PH 526
Year/Term
2026
Display other available years here.
Course Credits
4
Class meeting times
8 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

W1 D1: Math Bits
Introduction to the Course
Introduction to Central Forces
Class Notes OneNote Notebook GCF: Introduction to the Classical Central Force Problem
GCF: Introduction to the Quantum Central Force Problem
Unit: Math Bits (PDEs)
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Partial Differential Equations (Math Bits)
Unit: Math Bits (Power Series Solutions of ODEs)
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Power Series Solutions of ODEs (Math Bits)
W1 D2: Math Bits
Compare ISW to the Wave Equation
Compare ISW and Wave Equations
Eigenfunction Expansions
Optional, Advanced
GCF: Sturm-Liouville Theory
Video: Introduction to Sturm-Liouville Theory
Completeness
GCF: Completeness
Properties of Legendre Polynomials
Legendre Polynomials Lecture
GCF: Legendre Polynomials
GCF: Rodrigues' Formula
Wikipedia article on Legendre Polynomials
Laplace's Equation
W1 D3: Math Bits
Vector Spaces/Inner Products (Review)
Review Vector Spaces
GCF: Vector Spaces
Video: Vector Spaces
Video: Inner Products
HW 01 w/Solution
Legendre Polynomial Series
Guessing Legendre Coefficients
Expanding a Function in Legendre Polynomials.
GCF: Legendre Polynomial Series
GCF: Guessing and Visualizing a Legendre Polynomial Expansion
GCF: Calculating and Visualizing a Legendre Polynomial Expansion
W1 D4: Math Bits
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
Separation of Variables
Curvilinear
Separation of Variables - Spherical Harmonics
Video: Separation of Variables for a Particle Confined to a Sphere
GCF: Separation of Variables for the Hydrogen Atom
McIntyre 7.4
Legendre's Associated Equation
GCF: Associated Legendre Functions
McIntyre 7.6.2
Change of Variables (Review)
GCF: Change of Variables
Video: Change of Variables in Legendre's Equation
Series Solutions
Power Series Solution of ODEs: Math Example
GCF: Power Series Solutions: Method/Example
Theorems about Power Series Solutions
GCF: Power Series Solutions: Definitions and Theorems
Power Series Solutions
around Singular Points
(Frobenius Series)
W1 D5: Math Bits
Series Solutions for Legendre Polynomials
Series Solutions of Legendre's Equation
GCF: Series Solutions of Legendre's Equation
McIntyre 7.6.1
HW 02 w/Solution
W2 D1
PDEs in Physics
GCF: Important PDEs in Physics
GCF: Classification of PDEs
Video: Classification of PDEs
Separation of Variables
Rectangular (Review)
GCF: Separation of Variables
Particle in a 2-D Quantum Box
QuVis Simulation: 2-D Rectangular Box
QuVis Simulation: 2-D Circular Box
Quantum Cylinder
Quantum Cylinder
Boundary & Initial Conditions
GCF: Initial and Boundary Conditions on PDEs in Physics
GCF: Function Transformations
W2 D2
Systems of Particles
Newton's 2nd Law SWBQ
GCF: Systems of Particles
Taylor Chap. 3 (Optional)
Center of Mass
Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
The Two-Body Problem
Center of Mass-1
Center of Mass-2
GCF: Center of Mass
GCF: Reduced Mass
GCF: Central Forces
Taylor 8.1
Central Force Lagrangians
Reduced Mass
GCF: Reduced Mass
Taylor 8.2
W2 D3
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
HW 03 w/Solution
Angular Momentum
Angular Momentum SWBQ
Planar Orbit
Conservation of
Classical Angular Momentum
GCF: Angular Momentum
W2 D4
Velocity and Acceleration
in Polar Coordinates
Velocity and Acceleration in Polar Coordinates
GCF: Velocity and Acceleration
GCF: Derivatives of Basis Vectors
Acceleration in Polar Coordinates
Taylor 8.2
Review from PH 335:
Video: Time Derivatives of Polar Basis Vectors
Equations of Motion
GCF: Equations of Motion: \(\vec{F}=\mu\vec{a}\)
Taylor 8.3
Change of Variables
Polar Graphs
Circles, Ellipses, Hyperbolas
Polar Plots SWBQ
Conic Sections
GCF: Graphs in Polar Coordinates
Orbit Shape
GCF: Shape of the Orbit
Taylor 8.5
Energy and Angular Momentum
in Polar Coordinates
Angular Momentum in Polar Coordinates
GCF: Kinetic Energy and Angular Momentum
Spherical/Polar Coordinates
(Review)
Curvilinear Coordinates Introduction
Curvilinear Basis Vectors
GCF: Coordinates
Video: Basis Vectors (Review from PH 335/422)
W2 D5
Energy
GCF: Equations of Motion: Harmonic Oscillator
GCF: Equations of Motion: \(E=T+U\)
HW 04 Practice w/Solution
HW 04 w/Solution
W3 D1
More Effective Potentials
Effective Potentials
Effective Potentials
Acting Out Effective Potentials
Effective Potentials
Taylor 8.4 (Optional)
GCF: Effective Potentials
Inverse Square Forces:
Kepler's First Law
Keplerian Orbits and Effective Potentials
GCF: Effective Potentials w/ Orbits Taylor 8.6-8.7
Orbits for Inverse Square Forces
Kepler's First Law
Taylor 8.6-8.7
W3 D2
Kepler's Second Law
Kepler's Second Law
Effective Potential Diagrams, version 1
Free Particle Effective Potential
NASA: Orbits and Kepler's Laws
Scattering
Potentials
Air Hockey
GCF: Equations of Motion: Harmonic Oscillator
Orbits Summary
Classical Orbits Summary
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Classical Mechanics Orbits
GCF: Introduction to the Classical Central Force Problem
Unit: Quantum Mechanics on the Ring
Introduction to the Unit
Learning Outcomes
Unit Learning Outcomes: Quantum Mechanics on a Ring
GCF: Introduction to the Quantum Central Force Problem
W3 D3
Time Dependence Review
Time Evolution Refresher (Mini-Lecture)
Time Evolution of the ISW
HW 05 Practice w/Solution
HW 05 w/Solution
Reduced Mass in Quantum Systems
GCF: Reduced Mass
Introduction to Particle on a Ring
Intro to Particle on a Ring
GCF: Separation of Variables on a Ring
Separating out Center-of-Mass: Quantum
GCF: Reduced Mass
McIntyre 7.1
Eigenstates on a Quantum Ring
GCF: Motion on a Ring
McIntyre 7.5
W3 D4
Normalization of Eigenstates on the Ring
GCF: Normalization of States on a Ring
W3 D5
Probabilities for the Ring from Multiple Representations
Working with Representations on the Ring
GCF: Angular Momentum of the Particle on a Ring
Video: Matrix Representations for a Particle on a Ring
Video: Position Representations for a Particle on a Ring
Video: Degeneracy on the 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
GCF: Time Dependence of Ring States
GCF: Visualization of States on a Ring
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
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
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
GCF: Schrödinger's Equation in Spherical Coordinates
McIntyre 7.2
Rigid Rotor
GCF: Motion on a Sphere
McIntyre 7.4
McIntyre 7.6
Video: Separation of Variables for the Rigid Rotor
W4 D3
Spherical Harmonics:
Properties
Spherical Harmonics
GCF: Spherical Harmonics
McIntyre 7.6.4
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
Spherical Harmonics:
Visualization
Visualizing Spherical Harmonics
Visualizing Combinations of Spherical Harmonics
Spherical Harmonics on a Balloon
GCF: Visualizing Spherical Harmonics
Video: Description of Spherical Harmonics
McIntyre 7.6.5
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
GCF: The Radial Equation
W5 D3
Asymptotic Behavior
GCF: Asymptotic Solutions of the Radial Equation
W5 D4
Applications
Probability of Finding an Electron Inside the Bohr Radius
Classical Limit
Laguerre Polynomals
GCF: Power Series Solution of the Radial Equation
GCF: General Solution of the Radial Equation
Visualizing Radial Functions
Visualizing Radial Wave Functions
Video: Visualization of Radial Functions
Radial Functions
Radial Functions for a Hydrogenic Atom
Hydrogen Atom Wave Functions
McIntyre 8.5-8.6
GCF: States for the Hydrogen Atom
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
Visualization of Spherical Harmonics
Visualization of Hydrogen Atom
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
Learning outcomes
Unit Learning Outcomes: Partial Differential Equations (Math Bits)
Unit Learning Outcomes: Power Series Solutions of ODEs (Math Bits)
Unit Learning Outcomes: Classical Mechanics Orbits
Unit Learning Outcomes: Quantum Mechanics on a Ring
Unit Learning Outcomes: Angular Momentum in Quantum Mechanics
Unit Learning Outcomes: Quantum Mechanics of the (Unperturbed) Hydrogen Atom
Final Exam
FINAL EXAM