## Paradigms in Physics: Central Forces

Course name:
Course number:
PH 426
Course credits:
3
Class meeting times:
7 hours of lecture 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.

### Schedule from Spring 2022:

##### Homework

01 M 5/2
• One More Problem

Introduction

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)
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
(Review)
• 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

Potentials

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

Scattering
• 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

Midterm
• 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

Example:
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

Example:
Quantum Cylinder
• group Quantum Cylinder

16 M 5/23

Math Bits:
PDEs in Physics

Separation of Variables
Rectangular
(Review)

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

SchrÃ¶dinger'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:
Properties
McIntyre 7.6.4
• First Nine Terms
• Harmonics Verify P
• Spherical Harmonics Lookup

Spherical Harmonics:
Visualization
• 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
MEMORIAL DAY
No Class

22 T 5/31

McIntyre 8.1-8.4
Krane: Class Notes 3

Asymptotic Behavior
• Recurrence

Laguerre Polynomals

• computer Visualizing Radial Wave Functions

23 W 6/1

Hydrogen Atom Wave Functions
McIntyre 8.5-8.6

Calculations on Hydrogen Atom States
• 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

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

Vector Spaces
Inner Products

25 F 6/3

Review

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

26 M 6/6 Noon
FINAL EXAM