Central Forces

Course Credits

PH 426: Paradigms in Physics: Central Forces meets ~4 hours per week (TR at 10-11:50am) for 10 weeks for a total of 3 credits.

Prerequisites, Co-requisites and Enforced Prerequisites

Prereq: PH 213, PH 422, PH 425
Coreq: PH 335

Office Hours

(Note because of Covid-19, we are still working on our office hour assignments. This information may change.)

Weihong Qiu


Corinne Manogue (Math Bits)     corinne@physics.oregonstate.edu
Maggie Greenwood (TA)     greenwom@oregonstate.edu
Acacia Patterson (LA)     patterac@oregonstate.edu
Dustin Treece (LA)     treecedu@oregonstate.edu

Learning Resources

The required texts are McIntyre (McI), Taylor (T), and Manogue/Dray (MD).

Course Content

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. A brief introduction to classical scattering problems is also presented.  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 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 should 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
  • create a graph of the effective potential for systems with different potentials and use the graph to predict the behavior of the system
  • use several methods (including series solutions) to solve ordinary differential equations
  • 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.

Evaluation of Student Performance

(Note because of Covid-19, we are still working on our assessment plan. This information may change.)

  • 50% 10 required homework
  • 50% Final Exam.
  • Practice problems provide simple examples for you to check whether or not you understand the material as you go along.  They will not be graded.  Sometimes solutions will be posted.  At a minimum, you should read each practice problem and make sure that you know how to do it. If you can't, ask for help!
  • Required problems will be graded.  Solutions will be posted online.  Assignments turned in after solutions are posted can earn at most 50% of the total points.  Very late assignments will earn less.  It is a important to turn in what you have done by the due date, and, if necessary, the rest later. Please consult the instructor for special circumstances.
  • In 400/500 level classes, some of the required problems and probably one problem on the final exam will be marked as "Challenge" problems.   500-level students are required to do these Challenge problems.  400-level students are not necessarily expected to do them.  However, those students who hope to get an A are encouraged to do so.  While it may be possible for a 400-level student to get an A without doing any Challenge problems, it may be difficult.  (In PH 320, they are optional and don't count for anything--just fun.) Grading of the Challenge problems will be quite strict; we won't even look at them unless they seem to be clearly written, coherent, complete, and essentially correct.

Statement of Expectations for Student Conduct

Students will be expected to abide by all university rules regarding student conduct and academic honesty, in particular, see: link to University Rules.

Additional Ground Rules

Science is inherently a social and collaborative effort, each scientist building on the work of others. Nevertheless, each student must ultimately be responsible for his or her own education. Therefore, you will be expected to abide by a number of Ground Rules:

  1. We strongly encourage students to work with each other, more advanced students, the TA, and the professor, on assignments. However, each student is expected to turn in assignments that have been independently written up. In other words, the final synthesis must be entirely your own. This applies also to, and especially to, computer generated worksheets. If you work with someone on a computer project, do not get locked into writing the solution together. You will end up turning in the sameassignment.
  2. Homework solutions from previous years are very strictly off limits. You are on your honor not to use them, and not to share your homework solutions with other students. Allow faculty to use their time interacting with you, rather than continually thinking up new assignments. Besides, if you don't do the work yourself, it will show up very clearly on exams later. Likewise, the solutions are for your use only. You may make one copy and keep it in your personal files.
  3. Sources must be appropriately documented. If you find a homework problem worked out somewhere (other than homework solutions from previous years), you may certainly use that resource, just make sure you reference it properly. If someone else helps you solve a problem, reference that too. In a research paper, the appropriate reference would be:    Jane Doe, (private communication).
  4. Plagiarism – representing someone else’s work as your own – is unethical, but collaboration and exchange of ideas is healthy. You can avoid having collaborative efforts take on the look of plagiarism by acknowledging sources and by writing up your work independently.
  5. If you find that you have worked on a problem for 1/2 hour without making any forward progress, it would be a good idea to stop and seek help.

Statement Regarding Students with Disabilities

Accommodations for students with disabilities are determined and approved by Disability Access Services (DAS). If you, as a student, believe you are eligible for accommodations but have not obtained approval please contact DAS immediately at 541-737-4098 or at http://ds.oregonstate.edu. DAS notifies students and faculty members of approved academic accommodations and coordinates implementation of those accommodations. While not required, students and faculty members are encouraged to discuss details of the implementation of individual accommodations.