W1 D1: Math Bits

Compare ISW to the Wave Equation

Compare ISW and Wave Equations

Boundary & Initial Conditions

GCF: Initial and Boundary Conditions on PDEs in Physics
GCF: Function Transformations

Unit: Math Bits (PDEs)

Introduction to the Unit
Learning Outcomes

Unit Learning Outcomes: Partial Differential Equations (Math Bits)

W1 D2: Math Bits

Introduction to the Course

Introduction to Central Forces

GCF: Introduction to the Classical Central Force Problem
GCF: Introduction to the Quantum Central Force Problem

Properties of Legendre Polynomials

Legendre Polynomials Lecture

GCF: Legendre Polynomials
GCF: Rodrigues' Formula
Wikipedia article on Legendre Polynomials

Legendre Polynomial Series

Guessing Legendre Coefficients
Expanding a Function in Legendre Polynomials.

GCF: Legendre Polynomial Series
GCF: Calculating and Visualizing a Legendre Polynomial Expansion
GCF: Guessing and Visualizing a Legendre Polynomial Expansion

Sturm-Liouville Theory

GCF: Sturm-Liouville Theory
Video: Introduction to Sturm-Liouville Theory

W1 D3: Math Bits

Quantum vs. Classical Equations of Motion

HW 01 2025

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

Separation of Variables
Curvilinear

Separation of Variables on the Sphere

GCF: Separation of Variables for the Hydrogen Atom
McIntyre 7.4

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 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

Series Solutions

Power Series Solution of ODEs: Math Example

GCF: Power Series Solutions: Method/Example

Series Solutions for Legendre Polynomials

Series Solutions of Legendre's Equation

GCF: Series Solutions of Legendre's Equation
McIntyre 7.6.1

Theorems about Power Series Solutions

GCF: Power Series Solutions: Definitions and Theorems

Power Series Solutions
around Singular Points
(Frobenius Series)

W1 D5: Math Bits

Vector Spaces/Inner Products (Review)

Review Vector Spaces
GCF: Vector Spaces
Video: Vector Spaces
Video: Inner Products

HW 02 2025 w/Solution

Undoing Separation of Variables

Exponential Form of Fourier Series

GCF: Alternative Forms of Fourier Series
Video: Exponential Form of Fourier Series, aka Quantum States on a Ring

Particle in a 2-D Quantum Box

After Class:
QuVis Simulation: 2-D Rectangular Box
QuVis Simulation: 2-D Circular Box

Quantum Cylinder

Laplace's Equation

Completeness

GCF: Completeness

Lost Rows

Change of Variables (Review)

GCF: Change of Variables
Video: Change of Variables in Legendre's Equation

Legendre's Associated Equation

GCF: Associated Legendre Functions
McIntyre 7.6.2

W2 D1

Center of Mass

Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass

W2 D2

Systems of Particles

Newton's 2nd Law SWBQ

GCF: Systems of Particles
Taylor Chap. 3 (Optional)

HW 03 2025 w/Solution

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

Angular Momentum

Angular Momentum SWBQ

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
HW 04 w/Solution

More Effective Potentials

Effective Potentials

W3 D1

Effective Potentials

Acting Out Effective Potentials
Effective Potentials

Taylor 8.4 (Optional)
GCF: Effective Potentials

Orbits for Inverse Square Forces
Kepler's First Law

Taylor 8.6-8.7

W3 D2

Inverse Square Forces:
Kepler's First Law

Keplerian Orbits and Effective Potentials

GCF: Effective Potentials w/ Orbits Taylor 8.6-8.7

HW 05 w/Solution

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

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

HW 06 w/Solution

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

HW 07 w/Solution

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

HW 08 w/Solution

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

HW 09 w/Solution

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

HW 10 w/Solution

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

Topic/Day

Activities

Resources

Homework Due