- 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)

## Central Forces

The Central Forces Paradigm presents, in sequence, a classical and quantum mechanical treatment of the problem of two bodies moving under the influence of a mutual central force. The course begins with identifying this central force problem and reformulating the two-body problem in terms of a reduced mass. The classical part of this course asks the students to consider planetary orbits, emphasizing the use of energy and angular momentum conservation and an analysis of the effective potential. The quantum portion of course asks the students to find the analytic solution of the unperturbed hydrogen atom, which also includes an effective potential. This solution is built from simpler examples (a particle confined to a ring and a particle confined to a spherical shell) that introduce students to the relevant special functions needed for the full hydrogen atom solution.

The course also uses the paradigmatic example of a central force to introduce students to techniques for dealing with coupled differential equations, in particular breaking up a problem in several dimensions into problems involving one dimension at a time. In the classical part of the course, students use conserved quantities to break up a vector-valued ordinary differential equation into its spherical coordinate components. In the quantum part of the course, students use separation of variables to break the partial differential equation (Schrodinger's equation) up into single-coordinate eigenvalue equations.

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### Student Learning Outcomes

At the end of the course, students will be able to:

- characterize central forces and identify the similarities and differences between classical and quantum mechanics in the context of central forces
- discuss how conserved quantities (energy and angular momentum) constrain a physical system
- use several methods (including series solutions) to solve ordinary differential equations
- create a graph of the effective potential for systems with different potentials and use the graph to predict the behavior of the system
- use separation of variables to separate a partial differential equation into a set of ordinary differential equations
- for three different quantum systems: a particle confined to a ring, a particle confined to a spherical shell (rigid rotor), and the hydrogen atom,
- identify the Hamiltonian and energy eigenvalues for the given quantum system
- calculate probabilities, expectation values, uncertainties, and time evolution for the given quantum system

- use special functions to expand a generic quantum state in terms of the eigenfunctions of a complete set of commuting operators.

**Textbook:** Quantum Mechanics: A Paradigms Approach—-a textbook that follows the paradigms approach. The chapters that are relevant to the Central Forces course are: Ch 7: Angular Momentum and Ch 8:Hydrogen Atom

**Sample Syllabus:** Winter 2008

## Course 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)

### Activities Included

- All activities for Central Forces