## Physics 466, Fall 2017 Physical Mathematics

Welcome to physics 466 for 2017 .

We will be using a textbook called Physical Mathematics published by Cambridge University Press.

Insist on the 2014 printing which has many of the typos corrected.

The
videos of the lectures will be posted on YouTube:
lecture of 22 Aug 2017, sections 1.1-1.3.
lecture of 24 Aug 2017, sections 1.4-1.10.
lecture of 29 Aug 2017, sections 1.11-1.19.
lecture of 31 Aug 2017, sections 1.20-1.23.
lecture of 5 Sep 2017, sections 1.24-1.26.
lecture of 7 Sep 2017, sections 1.27-1.35. By mistake, the first half of the lecture was not recorded.
lecture of 12 Sep 2017, sections 1.36-1.38 and 2.1.
lecture of 14 Sep 2017, sections 2.1-2.6.
lecture of 19 Sep 2017, sections 2.7-2.10.
lecture of 21 Sep 2017, sections 2.13 and 3.1-3.3.
lecture of 26 Sep 2017, sections 3.4-3.10.
lecture of 28 Sep 2017, sections 3.11-3.13 and 4.1-4.4.
lecture of 3 Oct 2017, sections 4.5-4.14.
lecture of 5 Oct 2017, sections 4.14-16 & 5.1-6
lecture of 10 Oct 2017, sections 5.6-5.12.
lecture of 17 Oct 2017, sections 5.12-5.15.
lecture of 19 Oct 2017, sections 5.15-5.19.
lecture of 24 Oct 2017, sections 5.19-5.22.
lecture of 26 Oct 2017, sections 6.1--6.4.
lecture of 31 Oct 2017, sections 6.4--6.6.
lecture of 2 Nov 2017, sections 6.7--6.14.
lecture of 7 Nov 2017, sections 6.15--6.18.
lecture of 9 Nov 2017, sections 6.18--6.22 (Principle of stationary action, homogeneous first-order ordinary differential equations, linear first-order ordinary differential equations, small oscillations, singular points of second-order ordinary differential equations, method of Frobenius, Fuch's theorem).
lecture of 14 Nov 2017, sections 6.22--6.30 (Fuch's theorem, even and odd differential operators, Wronski's determinant, a second solution, why not three solutions? boundary conditions, a variational problem, formally self-adjoint differential operators, self-adjoint differential systems, making operators formally self adjoint).
lecture of 16 Nov 2017, sections 6.31--6.35 (making operators formally self adjoint, wronskians of self-adjoint operators, first-order self-adjoint differential operators, a constrained variational problem, eigenfunctions and eigenvalues of self-adjoint systems).
lecture of 21 Nov 2017: sections 6.35--6.44 and 8.1--8.3 (completeness of eigenfunctions of Sturm-Liouville systems, unboundedness of eigenvalues implies completeness, delta-function example, inequalities of Bessel and Schwarz, Green's functions, eigenfunctions and Green's functions, Green's functions in one dimension, principle of stationary action in field theory, symmetries and conserved quantities, systems of ordinary differential equations, nonlinear differential equations, Legendre polynomials, Rodrigues's formula, generating function for Legendre's polynomials).
lecture of 28 November 2017: sections 8.3--8.14 (generating function for Legendre's polynomials, Legendre's differential equation, recurrence relations, special values of Legendre's polynomials, Schlaefli's intgeral formula, orthogonal polynomials, azimuthally symmetric laplacian, Laplace's equation in two dimensions, Helmholtz's equation in spherical coordinates, associated Legendre functions, spherical harmonics, cosmic microwave background radiation).
lecture of 30 November 2017: sections 9.1--9.2 (Bessel functions of the first kind, spherical Bessel functions of the first kind, quantum dots).
lecture of 5 December 2017: sections 9.1--9.4 (Bessel functions of the first kind, wave-guides, spherical Bessel functions of the first kind, Bessel functions of the second kind, spherical Bessel functions of the second kind, scattering off a hard sphere).
lecture of 7 December 2017: sections 7.1--7.5 and 9.3--9.4 (turning differential equations into integral equations, Fredholm equations, Volterra equations, linearity, numerical solutions, integral transformations, Hankel functions, Bessel functions of the second kind, spherical bessel functions of the second kind).

ch2.pdf
Dirac on his delta function.pdf
Systems of ODEs: How to reduce a bunch of possibly high order and possibly time-dependent ODEs to an autonomous system of first-order ODEs and how to numerically integrate that system in Matlab and Python.
Hamilton systems and numerically integrating systems of differential equations (Chapter 15 of 2d edition)
Matlab codes for coupled harmonic oscilators, van der Pol oscillator, and Roessler system for c = 5.7

There is a list of errata at quantum.phys.unm.edu/466-15/errata.html. Please send new errata to me.

I started writing this book when Elsevier bought Academic Press the publisher of the book by Arfken et al. which I had been using.  Elsevier charges so much for its journals and books that it has made much of modern science inaccessible to all but the wealthiest institutions and individuals. When you start writing papers, you should post them on an arXiv (arxiv.org and/or bioRxiv.org) and submit them to journals not owned by Elsevier or Wiley. Pass it on.

Here is what I plan to cover in this course:
Linear algebra:             3 weeks
Fourier series:                1.5
Fourier transforms:      1.5
Infinite series:               1
Complex variables:       3
Differential equations:  3
Integral equations:        0.5
Legendre polynomials:  1.5
Bessel functions:            1.5

These topics are discussed in the first nine chapters of the book.

All homework problems are stated in the book
Physical Mathematics.

I will be doing some of the homework problems during the weekly problem session which is held on Wednesdays
at 2 pm in room 5.

Put homework in Changhao Yi's mailbox by 3:00 PM on its due date, usually a Friday. You can send him e-mail here.
You can send me e-mail
here.

Tentative list of homework assignments:

First homework assignment:  Do problems 1-14 of chapter 1 by Friday, 1 September.
Second homework assignment:  Do problems 15-28 of chapter 1 by Friday, 8 September.
Third homework assignment:  Do problems 29-40 of chapter 1 by Friday, 15 September.
Fourth homework assignment:  Do problems 1-14
of chapter 2 by Friday, 22 September.
Fifth homework assignment:  Do problems 15-23 of chapter 2
and 1-5 of chapter 3 by Friday, 29 September.
Sixth homework assignment:  Do problems 6-18 of chapter 3 and 1 of chapter 4 by Friday, 6 October.
Seventh homework assignment:  Do problems
2-18 of chapter 4 by Friday, 20 October.
Eighth homework assignment:  Do problems 19-22 of chapter 4
and 1-6 of chapter 5 by Friday, 27 October.
Ninth homework assignment:  Do problems 7, 10, 11, 13, 16, 17, 19, and 23 of chapter 5 by Friday, 3 November.
Tenth homework assignment:  Do problems 24-31 of chapter 5 by Friday, 10 November.
Eleventh homework assignment:  Do problems 32, 33, 36, 37, and 38 of chapter 5 and problems 1-5 of chapter 6 by Friday, 17 November.
List of homework problems of chapter 6 with fewer typos.
Twelfth homework assignment:  Do problems 6, 8-14, and 19 of chapter 6 by Friday, 1 December.
List of homework problems of chapter 8 with some new problems.
New version of chapter 9 with new problems.
Thirteenth homework assignment:  Using the above new set of exercises for chapter 8 and the new version of chapter 9 and its new exercises, do problems 12--15, 17, and 18 of chapter 8 and problems 9.1, 9.13, and 9.22 of the new chapter 9 by Friday, 8 December.

The final exam is on Thursday, 14 December in room 184 from 5:30 to 7:30.

The Trouble with Quantum Mechanics by Steven Weinberg. All students of physics should read at least section 1 of this essay.