Mathematical Methods

PHY 341

(Dated: August 21, 2004)

Dr. Perry Rice Room 13 Culler PHONE: 529-1374 E-MAIL ADDRESS: ricepr@muohio.edu WEB: http://www.cas.muohio.edu/ricepr/ Office Hours 10-11 M,T,TH or by appointment

Midterm 35%
Final 35%
Homework 30%

Texts: Essential Mathematical Methods for Physicists, Hans Weber and George Arfken. Also CRC Standard Mathematical Tables and Formulae. You will need some type of math table. This one is quite good, but you can use anyone you wish.

Prerequisites: Physics 181-2 (or 191-2), Phy 291-4, Intro. calculus sequence, linear algebra, curiosity about physics and willingness to work.

I view this course as the bridge between the introductory work of Phy 181-2, 291, and advanced courses in mechanics, electromagnetism, quantum and statistical physics. The course will deal with a variety of mathematical tools that are commonly used by physicists. Examples will be drawn from electromagnetism, quantum mechanics, and occasionally classical mechanics. The main purpose is to develop mathematical techniques for solving PHYSICS problems. There will be plenty of math of course, but we will try very hard to motivate in terms of applications in physics. This is not a math course, but a physics course. We will not be dealing with theorems and lemmas, but with electromagnetic fields and quantum states, and the math we have to use to deal with them. Experiments in the lab are the proof that our stuff works or not. Period!

Ch. 1 Vector Analysis (3 weeks) -Div, Grad, Curl, Gauss’ and Stokes theorems, potentials, differential and integral forms of Maxwell’s eqns., the wave equation, commutators

Ch. 2 Coordinate Systems (1 week) -transformations, emphasizing Cartesian, Spherical, and Cylindrical, separation of variables, Gauss law, Wave equation and Schroedinger equation

Ch 3-4 Determinants, Matrices, and Group Theory (3 weeks) -Hermitian matrices, eigenvalues, eigenvectors, soln’s to linear sets of differential equations, discrete and continuous groups (briefly), Coupled oscillators, Quantum mechanics of two and three level systems

Ch. 5 Infinite Series (1 day) -convergence tests, Taylor series, approximation techniques

Ch 8-9 Differential Equations and Sturm-Liouville Theory (2 weeks) -linear first and second order differential equations, and the generalized theory of orthogonal functions

Ch. 12 -Legendre Polynomials (2 weeks) -Applications of Sturm-Liouville theory to physics problems with a particular symmetry

Ch. 14 -Fourier Series and Transforms (4 weeks) -Periodic and non-periodic functions, solutions to differential equations.

Every day there will be a reading assignment, and/or a set of problems from Weber and Arfken. The reading assignment will be for the next day. I will assume you have done the reading, and will answer questions on it. If there are no questions I will assume you got it, and move on. Problems will be collected and a selection of those graded.

However, many exam questions will deal with those exact problems, or ones very similar. I will answer questions

about those problems, but will not provide you with a written out solution.

Exams will be open book (Arfken and Weber), open notes (your’s only), and you may use a math handbook.

Homework projects are extended calculations and/or Arfken and Weber problems that will be collected and graded.

Lectures will be brief, with lots of opportunity for questions. I will not just talk to show you how smart I may/may not be, or to hear myself talk. So if you have no questions, we may not meet very long at all on a given day. But I assure you that if you are honest with yourself, you will have questions

Questions are encouraged, in class or out.

Each student is expected to work COMPLETELY independently on all work. There are penalties for violations.

Late work is NOT ACCEPTED. If there is a problem, see me as early as possible, but unless you talk to me first, I will not accept late work.

Tests and homework are a form of communication, you are telling me what you know. I cannot read your mind! Solutions to problems should start with equations we have discussed in class or in the text, and a logical development should follow. Derivations found in other texts may be followed, but must be cast in terms of the lecture and/or text. If you have any questions about the above, please see me.

Do not fall behind. If you do not understand something, please see me. Spread your work out! Do not try and cram for this course! It will not work.

You will need to understand the material we will study to be effective in future physics courses. Also it will teach you how to think. I promise you that by the time we are done, your first two years of math and physics will be much more at your fingertips. So your basic skills will be well honed. And ultimately a boss will pay you for how much you can lift/carry/manufacture, or think, or some of each. They will not hire you, and say ”oh, problem 24 in chapter 4, thank heaven you’re here, we just can’t solve it”.

Some of the problems we will look at seem like oversimplifications of nature. Remember that physics is about modeling ultimately. If I am building a bridge, Newton will work fine. If I am building a nanoscale laser, well we need to deal with Schroedinger. If I want to explain the structure of the universe, I may need superstrings. We make predictions, and then do experiments. That’s what physicists do in the real world. Instead of philosophizing about how many teeth a horse may/must have, we open the mouth and count them.