## 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

This link will take you to the original version of the Central Forces page.

### Unit: Math Bits - Power Series Solutions to ODEs

**Hour 1: Math Bits - Series Solutions**- Summation Notation and Derivatives
- Solving a Differential Equation with Power Series

**Hour 2: Math Bits - Change of Variables**- Changing Independent Variables
- Changing Dependent Variables
- Optional: Solving the Hermite Equation

**Hour 3: Math Bits - Legendre's Equation**- Solving Legendre's Equation
- Introduction to Special Functions

### Unit: Classical Central Forces

**Hour 4: Systems of Particles**- Center of Mass
*Kinesthetic Activity: Survivor Outer Space*

- Reduced Mass

**Hour 5: Classical Angular Momentum**- Introduction to Angular Momentum
*Small Whiteboard Activity: Air Table*

**Hour 6: Kinematics in Polar Coordinates**- Polar Coordinates
- Vectors
*Small Group Activity: Velocity and Acceleration in Polar Coordinates*Add this activity

- Kepler's Second Law CAM added the next two bullets
- Angular Momentum and Kinetic Energy in Polar Coordinates

- Conic Sections
*Mathematica Activity: Plotting Conic Sections*CAM moved this activity

- Equations of Motion

- QUIZ
- Conservation and Orbital Shape

**Hours 9-10: Effective Potentials & Orbital Motion**- Effective Potentials
*Kinesthetic Activity: Interpreting Effective Potential Plots*

- Trajectories

**Hours 11-12: Scattering**- What Is Scattering?
- Cross-section Geometry
- Differential Cross Sections
*Small Group Activity: Hard Sphere*

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

- Review of Hamiltonians
- Separation of Variables

**Hours 14-15: QM States on a Ring**- QUIZ
- Energy Eigenstates for the Ring
- Angular Momentum for the Ring
*Small Group Activity: The Ring*

**Hour 16: Time Dependence on a Ring**- Energy Eigenstates, Superpositions, and Time Dependence
*Mathematica Activity: Visualizing Time Dependence*

### Unit: Math Bits - Boundary Value Problems

**Hours 17-18: Math Bits - Particle in a 2-D Box**- PDE Solution with 2 Space and 1 Time Variables
- Applying Boundary Conditions to PDE Solutions
*Small Group Activity: Infinite Square Well Squared*

- Degenerate Solutions to PDEs

**Hours 19-20: Math Bits - Legendre Polynomials**- Properties of Legendre Polynomials
*Mathematica Activity: Legendre Polynomial Expansions*

- Finding Legendre Series Coefficients

### Unit: The Quantum Rigid Rotor

- This topic needs more content

**Hour 22: Holiday**

**Hour 23: Spherical Harmonics**- Spherical Harmonics
*Kinesthetic Activity : Visualizing Spherical Harmonics*

**Hour 24-25: QM States for Rigid Rotor**- Spherical Harmonic Series
*Mathematica Activity : Plotting Spherical Harmonic Superpositions*

**Hour 26: Time Dependence for Rigid Rotor**- This topic needs new activities

- Review of Commutation
- Angular Momentum Operators

**Hour 28: Raising and Lowering Operators**

### Unit: The Hydrogen Atom

**Hour 29: The Radial Equation**- QUIZ
- Solving the Radial Wave Function
*Mathematica Activity: Visualizing Radial Wave Functions*

**Hours 30-31: The Hydrogen Atom****The Full Solution***Mathematica Activity: Visualizing Probability Density*

- Quantum Calculations

**Hour 32: The Classical Limit**- This topic needs more content

**Hours 33-34: Applications of Hydrogen Atom States**- Where Is the Electron?
- This topic needs more content

**Hour 35: Review**

### Activities Included

- All activities for Central Forces