Table of Contents

### Unit: Classical Central Forces

#### Center of Mass (35 minutes)

- Survivor Outer Space: A kinesthetic approach to introducing Center of Mass (Optional Kinesthetic Activity, 20 minutes)
- Derivation and Explanation of Center of Mass (Lecture, 15 minutes)

#### Introduction to Central Force Problems, Reduced Mass and Angular Momentum (1 hour 10 minutes)

- Assumptions about Central Force Motion (Lecture, 10 minutes)
- Derivation of Reduced Mass (Lecture, 20 minutes)
- Introduction of Angular Momentum (Lecture, 25 minutes)
- Definition of a Central Force (Lecture, 10 minutes)
- Central Forces on an Air Table (Small Whiteboard Activity, 15 minutes)

#### Polar Coordinates (40 minutes)

- Position Vectors in Polar Coordinates (Lecture/Discussion, 15 minutes)
- Plotting Conic Sections (Maple Activity, 25 minutes)

- Velocity and Acceleration in Polar Coordinates (Small Group Activity, 20 minutes)
- Kepler's 2nd Law in Polar Coordinates (Lecture, 5 minutes)

#### Solving the Central Force Equations of Motion for the Shape of the Orbit (90 minutes)

- Solution to the Central Force Equation of Motion (Lecture, 30 minutes)
- Finding the Shape of the Orbit (Lecture, 25 minutes)
- Finding the Radial Equation from Conservation of Energy (Lecture, 15 minutes)

#### Effective Potentials (90 minutes)

- Energy and Effective Potential (Lecture, 25 minutes)
- Exploring the Effective Potential (Maple Activity, 45 minutes)
- Interpreting Effective Potential Plots (Kinesthetic Activity, 15 minutes)
- Trajectories in an Attractive Central Potential (Maple/Java Activity, 30 minutes)

### Unit: Quantum Central Forces in One Dimension (The Ring Problem)

#### The Schrödinger's Equation for Central Forces (70 minutes)

- Review of Hamiltonians (Optional Lecture, 20 minutes)
- Derivation of the Hamiltonian in terms of the Reduced Mass (Optional Lecture, 20 minutes)
- Separation of Variables (Lecture, 30 minutes)

#### The Ring (2-3 hours)

- Finding the Eigenstates of Energy for the Ring (Lecture, 30 minutes)
- Angular Momentum for the Ring (Lecture, 20 minutes)
- Energy and Angular Momentum for a Particle Confined to a Ring (Small Group Activity, 30-90 minutes)
- Time Dependence for a Particle Confined to a Ring (Small Group Activity, 30 minutes)
- Visualizing the Probability Density for a Particle Confined to a Ring(Maple Activity, 30 minutes)
- Superposition States for a Particle Confined to a Ring (Optional Small Group Activity, 20 minutes)
- Expectation Values for a Particle Confined to a Ring (Optional Small Group Activity, 20 minutes)

### Unit: The Quantum Rigid Rotor

#### Solving the $\theta$ Equation for the Legendre Polynomial Series (2 hr)

This section may also be found in Math Bits.

- Solving the $\theta$ equations using a Series Solution Method (Lecture, 90 minutes)
- Guessing the Legendre Polynomial Expansion of a Function (Optional Maple Activity, 10-15 minutes)
- Legendre Series (Lecture, 20 minutes)
- Legendre Polynomial Series Coefficients (Maple Activity, 10-15 minutes)

#### Spherical Harmonics (3 hr)

- Associated Legendre Polynomials (Lecture, 20 minutes)
- Spherical Harmonics, the Solutions to the Rigid Rotor Problem (Lecture, 20 minutes)
- Visualizing Spherical Harmonics Using a Balloon (Kinesthetic, 30 minutes)
- Plotting the Spherical Harmonics (Maple Activity, 15 minutes)
- Combinations of $Y_{l,m}(\theta,\phi)$ and the Spherical Harmonic Series (Lecture, 25 minutes)
- Finding the Coefficients of a Spherical Harmonic Series (Small Group Activity, 25 minutes)
- Plotting Linear Combinations of Spherical Harmonics (Maple Activity, 15 minutes)
- Spherical Harmonics and the $H$, $L^2$, and $L_z$ Operators (Lecture, 60 minutes)

### Unit: The Hydrogen Atom

#### The Radial Equation (1 hr)

- Solving the Radial Equation (Lecture, 40 minutes)
- Visualizing Radial Wavefunctions (Maple Activity, 20 minutes)

#### The Hydrogen Atom (1 hr)

- Full Solutions to the Hydrogen Atom (Lecture, 45 minutes)
- Visualizing Hydrogen Probability Densities (Maple Activity, 20 minutes)
- Quantum Calculations on the Hydrogen Atom (Small Group Activity, 30 minutes)
- Probability of Finding an Electron Inside the Bohr Radius (Small Group Activity, 45 minutes)
- The Classical Limit (Lecture, 30 minutes)