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: Math Bits - Power Series Solutions to ODEs

• Summation Notation and Derivatives
• Solving a Differential Equation with Power Series
• Changing Independent Variables
• Changing Dependent Variables
• Optional: Solving the Hermite Equation
• Solving Legendre's Equation
• Introduction to Special Functions

Unit: Classical Central Forces

• Center of Mass
• Kinesthetic Activity: Survivor Outer Space
• Reduced Mass
• Introduction to Angular Momentum
• Small Whiteboard Activity: Air Table
• 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
• 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
• Energy Eigenstates, Superpositions, and Time Dependence
• Mathematica Activity: Visualizing Time Dependence

Unit: Math Bits - Boundary Value Problems

• 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
• Properties of Legendre Polynomials
• Mathematica Activity: Legendre Polynomial Expansions
• Finding Legendre Series Coefficients

Unit: The Quantum Rigid Rotor

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Hour 22: Holiday
• Spherical Harmonics
• Kinesthetic Activity : Visualizing Spherical Harmonics
• Spherical Harmonic Series
• Mathematica Activity : Plotting Spherical Harmonic Superpositions
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• Review of Commutation
• Angular Momentum Operators

Unit: The Hydrogen Atom

• 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
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• Where Is the Electron?
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Hour 35: Review

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