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.


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)

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

Polar Coordinates (40 minutes)

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

Effective Potentials (90 minutes)

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

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

The Ring (2-3 hours)

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.

Spherical Harmonics (3 hr)

Unit: The Hydrogen Atom

The Radial Equation (1 hr)

The Hydrogen Atom (1 hr)

Activities Included

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