Paradigms in Physics: Central Forces

Course name:
Paradigms in Physics: Central Forces
Course number:
PH 426
Course credits:
Class meeting times:
7 hours of lecture per week for five weeks
PH213, PH425, PH422
Course description:
Gravitational and electrostatic forces; angular momentum and spherical harmonics, separation of variables in classical and quantum mechanics, hydrogen atom.

Schedule from Spring 2022:


01 M 5/2
  • One More Problem


Center of Mass

Systems of Particles
Taylor Chap. 3
Taylor 8.3, first part

02 T 5/3

The Two-Body Problem
Taylor 8.1
  • Central Force Definition
  • Central Force with no External Forces
  • Conservative Forces

Reduced Mass
  • face Reduced Mass Lecture Notes
Taylor 8.2
  • Center of Mass for Two Uncoupled Particles
  • Reduced Mass
  • Sun vs. Jupiter
  • Undo Formulas for Reduced Mass (Algebra)
  • Undo Formulas for Reduced Mass (Geometry)

Spherical/Polar Coordinates
Review from 335:
Video: Basis Vectors
  • Angular Momentum and Kinetic Energy in the Center of Mass
  • Polar Basis Vectors
  • Polar vs. Spherical Coordinates
  • Positions and Velocities in Curvilinear Coordinates

Velocity and Acceleration
  • group Velocity and Acceleration in Polar Coordinates
  • face Acceleration in Polar Coordinates

03 W 5/4

Polar Graphs
Circles, Ellipses, Hyperbolas
  • group Conic Sections
  • Eccentricity
  • Lines in Polar Coordinates

04 Th 5/5

Conservation of
Classical Angular Momentum
  • face Conservation of Angular Momentum
  • Central Forces are Conservative

Equations of Motion: F=ma
  • face Equations of Motion Lecture Notes
Taylor 8.3, second part
  • Newtonian Mechanics: Central Force
  • WQ-Lagrangian Mechanics for a Central Force

Change of Variables

Orbit Shape
  • face Orbit Shape
  • Find Force Law: Logarithmic Spiral Orbit
  • Find Force Law: Spiral Orbit

Orbits for Inverse Square Forces
Taylor 8.6-8.7

05 F 5/6


Equations of Motion: E=T+U
  • face Equations of Motion from Energy Conservation

Effective Potentials
  • Hockey
  • NASA
  • NASA 2
  • Scattering
  • WQ-Effective Potentials
  • Yukawa
  • Yukawa 2

Inverse Square Forces:
Kepler's First Law
  • group Keplerian Orbits

Orbits Summary
  • face Classical Orbits Summary

06 M 5/9

  • Hard Cone Scattering
  • Solid Angle

07 T 5/10

Eigenstates on a Quantum Ring
Course Notes 14
McIntyre 7.5

08 W 5/11

Probabilities for the Ring from Multiple Representations
  • Activity: Representations of Probabilities on the Ring
  • Activity: Working with Representations on the Ring
  • Energy and Angular Momentum for a Quantum Particle on a Ring Handout
  • Normalization of Quantum States
  • QM Ring Compare
  • QM Ring Function
  • QMRingCompareSeparated
  • Representations of Probabilities on the Ring
  • Ring Function
  • Ring Ket Book
  • Ring Table
  • Working with Representations on the Ring

09 Th 5/12

Meaning of the Wave Function
  • Probability Density Ring

10 F 5/13

Time Dependence Review

Time Dependence on a Ring
  • Expectation Values for a Particle on a Ring Handout
  • Particle on a Ring Time Dependence: Mathematica Notebook
  • QM Ring Function with Time Dependence
  • QM Ring Time
  • QM Ring with Time Dependence
  • QMRing_TimeDep_Intro
  • Ring Ket
  • Superposition States for a Particle on a Ring
  • Time Dependence for a Quantum Particle on a Ring
  • Visualization of Wave Functions on a Ring
  • WQ_Particle_On_The_Ring_I

11 M 5/16

  • face Classical Orbits Equation Sheet
  • face Quantum Ring Equation Sheet
  • face Classical Orbits Summary

12 T 5/17

To prepare for class:
Review Power Series
Power Series Review
Review Power Series
GMM: Power Series

Math Bits:
Series Solutions
  • group Power Series Solution of ODEs: Math Example
  • Hermite Polynomials
  • Laguerre
  • Laguerre Polynomials
  • Laguerre Polynomials
  • Legendre
  • ODE Power Series Solutions One
  • ODE Power Series Solutions Two
  • Power Series Solution for Laguerre Polynomials
  • Recurrence Math
  • Recurrence Relations
  • Sum Shift

13 W 5/18

Power Series Theorems

Series Solutions for Legendre Polynomials
  • face Series Solutions of Legendre's Equation
McIntyre 7.6.1

Properties of Legendre Polynomials
  • face Legendre Polynomials Lecture
  • Legendre Polynomial Look Up
  • Properties of Legendre Polynomials

14 Th 5/19

Math Bits:
Sturm-Liouville Theory
  • face Sturm-Liouville Theorems

Math Bits:
Legendre Polynomial Series
  • computer Guessing Legendre Coefficients
  • face Legendre Polynomial Series Lecture
  • computer Expanding a Function in Legendre Polynomials.
  • Legendre Polynomial Series for the Sine Function

15 F 5/20

Particle in a 2-D Quantum Box
  • Laplace
  • Laplace Practice
  • Quantum Particle in a 2-D Box
  • Quantum Particle in a 2D Box with Time Dependence
  • Quantum Particle in a 2D Box: Complete
  • Square 2D
  • Square 2D Fourier

Quantum Cylinder
  • group Quantum Cylinder

16 M 5/23

Math Bits:
PDEs in Physics

Separation of Variables

Math Bits:
Separation of Variables in Curvilinear Coordinates
  • Laplace's Equation in Polar Coordinates
  • Separation of Variables on the Sphere

17 T 5/24

Separating out Center-of-Mass: Quantum
McIntyre 7.1

Schrödinger's Equation:
Spherical Symmetry
Course Notes 10-13
McIntyre 7.2

Rigid Rotor
McIntyre 7.4
McIntyre 7.6
Video: Separation of Variables for the Rigid Rotor
Course Notes 15-17

Review: Change of Variables
  • ODE Change of Independent Variable

Legendre's Associated Equation
McIntyre 7.6.2
  • Associated Legendre

Spherical Harmonics:
McIntyre 7.6.4
  • First Nine Terms
  • Harmonics Verify P
  • Spherical Harmonics Lookup

Spherical Harmonics:
  • Quantum State Visualization
  • Visualization of a Particle on a Sphere Superpositions

Symmetries, Parity, Degeneracy

18 W 5/25

Quantum States of the Rigid Rotor
  • face Quantum Reference Sheet
  • group Probabilities for a Quantum Particle on a Unit Sphere in Spherical Harmonic Functions
  • group Finding Coefficients of a Spherical Harmonics Series
  • Angular Momentum Probability
  • Quantum Numbers on the Sphere
  • Rigid Rotor Probabililties Practice
  • Sphere
  • Sphere Questions
  • Sphere Questions: Matrix Notation
  • Sphere Table
  • Spherical Harmonics Expansion

19 Th 5/26

Orbital Angular Momentum:
Spin 1 revisited
McIntyre 7.3

Commutation Relations
  • Angular Momentum Commutation Relations
  • WQ_Angular_Momentum_Algebra_I

Raising and Lowering Operators
Griffiths 4.3
  • Raising and Lowering Operators for Spin

20 F 5/27

Time Dependence
  • group Quantum Particle on a Unit Sphere in Dirac Notation
McIntyre 7.6.3
  • QM Sphere with Time Dependence

Math Bits:
Boundary & Initial Conditions
  • Quantum Cylinder

21 M 5/30
No Class

22 T 5/31

The Radial Equation
McIntyre 8.1-8.4
Krane: Class Notes 3

Asymptotic Behavior
  • Recurrence

Laguerre Polynomals

Radial Functions

Visualizing Radial Functions
  • computer Visualizing Radial Wave Functions

23 W 6/1

Hydrogen Atom Wave Functions
McIntyre 8.5-8.6

Calculations on Hydrogen Atom States
  • Bohr Radius
  • Eigenvalues for Different Systems
  • Hydrogen Atom Representation Matching
  • Hydrogen Atom Table
  • Hydrogen, Version 1
  • Hydrogen, Version 2
  • Probability Density from the Density Operator
  • SP Hybrid
  • SP not Hybrid
  • SPHybrid2
  • Verify Eigenvalues for a Hydrogen Atom State

24 Th 6/2

Hydrogen Spectra
McIntyre 8.3

Symmetries, Parity, Degeneracy

Hydrogen Atom Visualization of Probability Densities
  • Interpreting Hydrogen Atom Plots

Classical Limit

  • group Probability of Finding an Electron Inside the Bohr Radius
  • Dipole Moment
  • Forbidden Region

Vector Spaces
Inner Products

25 F 6/3


Midterm Revisit
  • Air Table
  • Effective Potentials: Graphical Version
  • Probability Density

26 M 6/6 Noon