Table of Contents

# PH426: Paradigms in Physics, Central Forces

## Winter 2011

This course is taught by Prof. Ethan Minot during weeks weeks 8, 9 and 10 of Winter Quarter 2011. “Central Forces” is the 6th part of a 9 part sequence called Paradigms in Physics.

## Calendar

Day | Topic | Summary | Additional Reading | Assignments | ||
---|---|---|---|---|---|---|

1* | M | Two-body Problems | Classical Angular Momentum | week1notes.pdf | Start hw1 | |

2 | Tu | Describing trajectories in polar coordinates | worksheet: polar acceleration, worksheet: conic sections, Maple9: conic sections | |||

3 | W | Equations of motion | hw1soln.pdf | |||

4 | Th | Orbit shapes | Worksheet: Orbits and Maple14: Orbits | Start hw2 | ||

5 | F | Inverse square force laws | practice_questions_for_final.doc | |||

Day | Topic | Summary | Additional Reading | Assignments | ||

6 | M | Schrödinger's Eqn | Recent Expt: Buckyball interference pattern | QM course packet 7-1 to 7-7 | hw2soln.pdf. Start hw3 | |

7 | Tu | Quantum States on a Ring | Worksheet: QM Ring | 7-7 to 7-28 | ||

8* | W | Time Dependence on a Ring | Maple14: QM Ring, Recent Expt: Electrons in a metal ring | hw3soln.pdf, Start hw4 | ||

9* | Th | Legendre's Equation | Maple14: Guess Legendre, Maple11: Find Legendre Coefficients | |||

10* | F | Spherical Harmonics | hw4soln | |||

Day | Topic | Summary | Additional Reading | Assignments | ||

11 | M | Spherical Harmonics | Maple14: Plot spherical harmonics, | Start hw5 | ||

12 | Tu | Quantum States of the Rigid Rotor | Maple14: YLM Combinations | |||

13 | W | Radial Functions | Maple14: Radial wavefn | hw5soln.pdf, Start hw6 | ||

14 | Th | Hydrogen Atom Wave Functions | Maple14: Hydrogen Visualization | |||

15 | F | The Classical Limit + Review | Practice questions for final | hw6soln |

**Classes on these days will be run by Corinne Manogue, Ethan Minot will be travelling.*

## Syllabus

##### Classical Mechanics

- Systems of Particles
- Center of Mass

- Two-body Problems
- Central Forces
- Reduced Mass

- Classical Angular Momentum
- Torque
- Conservation of Angular Momentum
- Kepler's Second Law

- Spherical Coordinates
- Velocity and Acceleration in polar coordinates
- Circles, Ellipses, Hyperbolas
- Conic Sections

- Equations of Motion:
*F*=*ma* - Orbit Shapes
- Inverse Square Forces
- Kepler's first law

- Equations of Motion:
*E*=*T*+*U* - Effective Potentials
- Other Force Laws

##### Quantum Mechanics

- Schrödinger's Eqn.
- Spherical Symmetry
- Separation of Variables

- The Ring
- z-component of angular momentum
- Degeneracy
- Comparison to Particle in a Box

- Quantum States on a Ring
- Time Dependence on a Ring

- The Rigid Rotor
- Legendre's Equation
- Change of Variables
- Series Solutions
- Legendre Polynomials
- Rodrigues' formula
- Graphs
- Orthonormality
- Legendre Polynomial Series
- Legendre's Associated Equation
- Legendre Functions

- Spherical Harmonics
- Orthonormality

- Spherical Harmonics
- YLM's
- Symmetries, parity, degeneracy

- Quantum States of the Rigid Rotor
- Linear Combinations of YLMs

- Angular Momentum
- Commutation Relations

- Orbital Angular Momentum
- Raising & lowering operators
- Spin 1 revisited

- The Radial Equation
- Radial Functions
- Asymptotic Behavior
- Laguerre polynomials

- Hydrogen Atom Wave Functions
- Symmetries, parity, degeneracy
- Probablility Densities

- Applications
- The Classical Limit