# Physics 466 Mathematical Methods of Theoretical Physics

Mathematical Methods for Theoretical Physics (466-001)

Fall 2005 Tuesdays and Thursdays from 17:30 to 18:45 in room 5. My e-mail address is cahill@unm.edu. My office is room 176, and my phone number is 277-5318. My office hours are by appointment. But in fact whenever you see me, I am available for questions about this course.

The grader is Mr. Tianjian Lu . His phone number is 254-5308. His e-mail address is tjlu@unm.edu.

A pdf file on linear algebra, Fourier series, Fourier transforms, series, complex-variable theory, and tensors is available as the file nascent book.

Some notes of mine on differential equations appear in these pdf files: pages 340-368, pages 369-412, pages 413-433, pages 434-448, pages 449-463.

First homework assignment: Do the problem assigned in class as well as these from the 5th edition of Arfken & Weber: 1.1.11, 1.3.4, 1.4.1, 1.4.16, 1.5.5, 1.5.6, 1.5.18. Eq.(12.168) is a formula for the dot-product of two unit vectors n_1 & n_2 with polar angles theta_1, phi_1 and theta_2, phi_2:
cos(n_1, n_2) = n_1 . n_2 = cos(theta_1) cos(theta_2) + sin(theta_1) sin(theta_2) cos(phi_1 - phi_2).
This first homework assignment is due in class on Tuesday 9/13/05.

Here are the solutions to the first homework assignment.

Second homework assignment: Do problems 2-5 listed at the end of chapter one of the nascent book. Lapack is one way to do problems 4 & 5, but students may use Matlab or Maple or any other program. Those who want Lapack on their PCs can get it here, where they will find ways to download linux rpms. Also, do problems 3.1.2, 3.2.(2,7,9,13,14,15,16a), 3.3.12, 3.4.(7,9), 3.6.20. This assignment due on 10/6.

Here are the solutions to most of the problems of the second homework assignment. Here is the solution to problem 1.3. Problem 1.2 is done in the nascent book.

Third homework assignment: Do 14.1.(1,3,8); 14.3.5 & 14.4.2; and 14.4.(9,10,11) by 20 October.
Equations(14.11 and 14.12) of Arfken are the same as Eqs.(2.30) & (2.31) of the on-line notes/book.

Here are the solutions to the problems of the third homework assignment.

Fourth homework assignment: Do problems 15.3.(6, 10, 16); 15.4.(1, 3, 4) & 6.1.1; and 6.2.1 by 3 November.

Here are the solutions to the problems of the fourth homework assignment.

Fifth homework assignment: Do problems 6.2.4; 6.4.4; 6.5.8; 6.7.1; 7.1.4a; 7.2.(5, 7 for a > b > 0, 14b, 16) by 17 November.

Here are the solutions to the problems of the fifth homework assignment.

Sixth homework assignment: Do problems 8.2.14, 8.3.5, 8.5.5, 8.6.(9 & 15), 9.1.4, 9.2.5, 12.3.15, and 12.5.15 by 8 December. Here is Eq.(8.83), which problem 8.6.15 refers to. Here is section 12.5, which problem 12.5.15 refers to. In doing problem 12.3.15, students might want to refer to the binomial theorem discussed at the end of chapter 4 of the on-line book.

Here are the solutions to the problems of the sixth homework assignment.

Seventh homework assignment, a.k.a., the take-home exam (due Friday 12/16): Do problems 14.3.(6, 7), 11.7.27, and these special problems:

1. A neutron at rest has a mean lifetime of 885.7 s. What is the mean lifetime of a neutron whose energy is 100 times its rest-mass energy?
2. A green laser beam is shot vertically from the surface of the Earth and received by a detector that is in a geostationary orbit of radius 35,786 km in the plane of the equator. (I gave credit for both answers.) Find the dimensionless gravitational potential phi = GM/(r c^2) on the surface of the Earth and on the receiving satellite, and compute the change in the frequency of the laser light if its frequency is 5.64 x 10^{14} Hz.
3. How big must a sphere of chocolate ice-cream be if it is to be a black hole? You may take the density of the ice-cream to be that of water at standard temperature and pressure, to wit, 1 gram per cubic centimeter.
4. In doing problems 14.3.(6, 7), the (new) section 8.7 of the on-line notes might be helpful.

In problem 11.7.27, you may want to write a computer program to find the first zero of the derivative of each spherical Bessel function. Do not bother to discuss why it is more efficient to work with the derivative rather than the function itself.

Here are the solutions to the first three and the second three problems of the take-home exam.