Course Name
Paradigms in Physics: Central Forces
Course Number
ph426
Year/Term
Spring-2024
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
W1 D1
Center of Mass
Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Systems of Particles
Taylor Chap. 3 (Optional)
W1 D2
Angular Momentum
Conservation of
Classical Angular Momentum
Conservation of Angular Momentum Lecture Notes
The Two-Body Problem
Taylor 8.1
Equations of Motion
Taylor 8.3
Spherical/Polar Coordinates
(Review)
Curvilinear Coordinates Introduction
Video: Basis Vectors (Review from PH 335/422)
Velocity, Acceleration, and Angular Momentum
in Polar Coordinates
Central Force Lagrangians
Reduced Mass
Taylor 8.2
W1 D3
Energy
Effective Potentials
Acting Out Effective Potentials Effective Potentials
Taylor 8.4 (Optional)
GMM: Effective Potentials
W1 D4
More Effective Potentials
Potentials
Solutions of the Harmonic Oscillator Equation (Review)
Orbit Shape
Taylor 8.5
W1 D5
Polar Graphs
Circles, Ellipses, Hyperbolas
W2 D1
Inverse Square Forces:
Kepler's First Law
W2 D2
Kepler's Second Law
Orbits Summary
Unit: Quantum Mechanics on the Ring
Introduction to the Unit
Learning Outcomes
W2 D3
Introduction to Particle on a Ring
Time Dependence Review
Eigenstates on a Quantum Ring
Course Notes 16
McIntyre 7.5
W2 D4
Meaning of the Wave Function
Normalization of Eigenstates on the Ring
Course Notes 16
W2 D5
Time Dependence on a Ring
Unit: Math Bits (Power Series Solutions of ODEs)
Introduction to the Unit
Learning Outcomes
W3 D1
W3 D2
Math Bits: Series Solutions for Legendre Polynomials
Course Notes 21
McIntyre 7.6.1
Math Bits: Theorems about Power Series Solutions
Math Bits: Power Series Solutions
around Singular Points
(Frobenius Series)
Math Bits: Properties of Legendre Polynomials
Math Bits: Legendre's Associated Equation
McIntyre 7.6.2
Unit: Math Bits (PDEs)
Introduction to the Unit
Learning Outcomes
W3 D3
Math Bits: Vector Spaces (Review)
Before Class:
Review Vector Spaces
GMM: Definition of a Vector Space
Video: Vector Spaces
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
Math Bits Example:
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:
Boundary & Initial Conditions
Math Bits:
Legendre Polynomial Series
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
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)
Rigid Rotor
Course Notes 25
McIntyre 7.4
McIntyre 7.6
Video: Separation of Variables for the Rigid Rotor
W4 D3
Spherical Harmonics:
Properties
McIntyre 7.6.4
Spherical Harmonics:
Visualization
W4 D4
Quantum States of the Rigid Rotor
Spherical Harmonics on a Balloon Quantum Systems Reference Sheet
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
McIntyre 7.6.3
Unit: Quantum Mechanics of the Hydrogen Atom
W5 D2
Commutation Relations
The Radial Equation
McIntyre 8.1-8.4
Krane: Class Notes 3
W5 D3
Asymptotic Behavior
W5 D4
Radial Functions
Visualizing Radial Functions
Laguerre Polynomals
Hydrogen Atom Wave Functions
McIntyre 8.5-8.6
Calculations on Hydrogen Atom States
Hydrogen Spectra
McIntyre 8.3
Hydrogen Atom Visualization of Probability Densities
W5 D5
Final Exam
FINAL EXAM