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PHYS 320: Mathematical Physics

Seth Major

In this course we will review, learn, and practice a number of mathematical tools which have practical use for physicists. The idea will be that these tools will be of use, not only in your remaining studies at Hamilton, but also in graduate school.

A "pedestrian guide'' to the mathematics, the course focus on the implementation of the methods and applications rather than proofs and placing the results in context within mathematics. So, even if you have had a course in one or more of the topics, the emphasis and even some of the methods will be new. I also hope that you will master much of the material so that you can easily use in research, further studies, or elsewhere in life.

The course is in lecture/discussion format punctuated by short presentations. I strongly encourage folks to ask questions and make observations. Much of the class time is spent making connections to physics. The course is somewhat unusual in that one first encounters new material in the reading and in a small number of problems. In class we focus on filling out understanding, answering questions, embellishing the material, and working through more examples.

Course information (pdf):

Course syllabus & info

Special functions presentation info

Questions, Problems and Reading (pdf):

Questions 1
Questions 2
Questions 3
Questions 4
Questions 5
Questions 6
Questions 7
PS 1
Questions 8
Questions 9
Questions 10
Questions 11
Questions 12
PS 2
Questions 13
Questions 14
Questions 15
Questions 16
PS 3
Questions 17
PS 4

Some recent topics:

  • Complex Numbers

  • Methods of solving Ordinary Diff. Equ's (ODEs)

    First order, linear second order equations, Laplace transformations, Green functions, series method, ...

  • Sturm-Liouville Theory

    The remarkable properties of solutions to self-adjount differential equations

  • Special Functions

    Gamma, Legendre, Hermite, Bessels, Laguerre, Chebyshev even Hypergeometric functions will be our delight

  • Fourier Series & Transforms

    A theorem, computation of coefficients, Fourier transform, Dirac delta-function

  • Partial Diff Equ's (PDEs)

    Waves, diffusion, Schrodinger, Laplace, separation of variables

  • Techniques of Complex Analysis

    Complex number review, analytic functions, integration, integral theorems

  • Tensors

    (Notes) definition and applications

  • Groups

    Introduction, examples, rotations in 3-space, representations, applications

Handy links:

The NIST Digital Library of Mathematical Functions.

Referenced links:

Curious to see more on the energy discussion we started the semester with? I started with a article in Physics Today (July 2016).

Michael Berry's Physics Today article on Special Functions.

A video of standing waves on a soap film disk. The Bessel functions are identified. The background music may be safely ignored.

Reading (3.9 MB pdf) on Sturm-Liouville theory from Arfken and Weber.

Reading on the Frobenius method from Arfken. An example of the method starts on page 395.

The Bessel-y initial condition algebra for the "linearly lengthening pendulum".

What do 3D parabolic coordinates look like?

Reading on vector differentiation, particularly time dependent basis vectors (from Potter and Goldberg)

Dirac Notation Primer (There is also a little bit on the notation in Boas, see page 181.)

Power spectrum (CMBR) Planck including polarization data, WMAP (5 yr results). The black body spectrum from COBE which helped win the 2006 Nobel Prize

BBC news on the Earth's geoid. Here's a page with the expansion of the Earth's gravitational potential in terms of associated Legendre polynominals.

Lovely Simulations of hydrogenic orbitals from

More on the 1938 Christmas discovery of Meitner and Frisch. Frisch visited his aunt who was in exile in Sweden.

Phet fission demo

The primer.

The bare bones mathematica notebook on the double pendulum.

For more on spin networks you can dip into the spin network primer.


Math Methods Poetry (pdf):

Spring 2009 Creations


Last modified 5 May 2017 Link to Seth's Net Home Link to Department of Physics link to archives link to gr-qc link to gr-qc/new link to archive form