Instructor: Kevin Cahill

Phone: 505 205 5448, but please call only after 1 PM.

Due to the pandemic, all classes will be held

online via Zoom from 23 August

except for the week of 25 September.

The Zoom link is https://unm.zoom.us/j/9119790161.

Meeting ID: 911 979 0161

Passcode: class

I plan to use email to send you invitations for each class

but you may find it more convenient just to click on the

Zoom link https://unm.zoom.us/j/9119790161.

466 Classes are at 5:00 on Tuesdays and Thursdays.

468 Classes are at 5:30 on Wednesdays.

Office hour: Mondays at 3 o'clock in the courtyard of PAIS.

468 is part of 466; students should attend 468

even if they are not registered for the class.

The plan is to cover the first 10 chapters of the book and also some of the most interesting parts of the second half of the book.

We will be using the second edition of a textbook called

New and used copies of the book are available in the UNM bookstore, but you can also get it from Amazon and eBooks.

Errata for 2d edition

Class notes. This file is frequently updated.

Handwritten class notes from iPad. (Please skip the first blank page.)

Notes on linear algebra, vector calculus, Fourier series, Fourier and Laplace transforms, and infinite series.

Handwritten class notes from iPad. (Please skip the first blank page.)

6.1 Analytic Functions, 6.2 Cauchy-Riemann Conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula, 6.5 Harmonic Functions, 6.6 Taylor Series for Analytic Functions, 6.7 Cauchy’s Inequality, 6.8 Liouville’s Theorem, 6.9 Fundamental Theorem of Algebra, 6.10 Laurent Series, 6.11 Singularities, 6.12 Analytic Continuation, 6.13 Calculus of residues, 6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots, 6.17 Conformal mapping, 6.18 Cauchy’s principal value, 6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, 6.22 Method of Steepest Descent, 6.23 Applications to string theory

GRADING — Grading will be based 100 percent on homework.

HOMEWORK — Send to grader via email.

Due Monday 12 September 2022:

do problems 18, 19 28, 32, 33, and 34 of chapter 1. Here problem 34 is a new problem (1.34) described in the online class notes.

Due Monday 19 September:

Do problems 2.1, 2.7, 2.8, 3.19, 3.25.

Due Monday 26 September:

Do problems 4.6, 4.7, 4.18, 5.1, 5.2, 5.12.

Due Monday 3 October:

Do problems 5.19, 6.1, 6.5, 6.7, 6.11.

Due Monday 10 October:

Do problems 6.24, 6.25, 6.26, 6.29, 6.34

Due Monday 17 October:

Do problems 7.9, 7.10, 7.11, 7.13.

VIDEOS of classes:

Google first displays the videos in SD and then hours later in HD.

So if the images are fuzzy, try them again the next day. If they're still fuzzy, send me email.

23 Aug Tuesday

Arrays, matrices, linear operators, inner products, inequalities, linear independence, dimension, orthonormal vectors, and Dirac notation.

24 Aug Wednesday

Dirac notation, adjoints of operators, hermitian and unitary operators, Hilbert spaces, determinants, jacobians, linear least squares, and Lagrange multipliers.

25 Aug Thursday

Eigenvectors and eigenvalues of square matrices, the characteristic equation, functions of matrices, hermitian and normal matrices, compatible matrices, the singular-value decomposition, pseudoinverses, and entanglement.

30 Aug Tuesday

Singular-value decomposition, derivatives, complex differentiation, gradient, divergence, laplacian, curl, Helmholtz decomposition, Dirac's delta function, covariant and contravariant vectors.

31 Aug Wednesday

Fourier series, the interval, where to put the 2pi's, real Fourier series, stretched intervals, Fourier series for functions of several variables, integration and differentiation of Fourier series, convergence of Fourier series.

1 Sep Thursday

Sections 3.8 How Fourier Series Converge, 3.9 Measure and Lebesgue Integration, 3.10 Quantum-Mechanical Examples, 3.11 Dirac’s Delta Function, 3.12 Nonrelativistic Strings, 3.13 Periodic Boundary Conditions, 4.1 Fourier transforms.

Due to FERPA, I have edited out any segments in which a student's voice (or image) was recorded. To avoid FERPA problems, and since I am hard of hearing, and because some of us have English as a second, third, or fourth language, please use Zoom's chat instead of speaking -- at least until I have stopped recording the lecture.

6 Sep Tuesday

Sections 4.1 Fourier transforms, 4.2 Fourier transforms of real functions, 4.3 Dirac, Parseval, and Poisson, 4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables.

7 Sep Wednesday

4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables, 4.6 Convolutions, 4.7 Fourier transform of a convolution, 4.8 Fourier transforms and Green’s functions, 4.9 Laplace transforms, 4.10 Inversion of Laplace transforms.

8 Sep Thursday

4.10 Inversion of Laplace transforms, 4.11 Volterra’s Convolution, 4.12 Derivatives and integrals of Laplace transforms, 4.13 Laplace transforms and differential equations, 4.14 Applications to Differential Equations, 5.1 Convergence, 5.2 Tests of convergence, 5.3 Convergent series of functions, 5.4 Power series, 5.5 Factorials and the gamma function

13 Sep Tuesday

5.5 Factorials and the Gamma Function, 5.6 Exponential Integral and Incomplete Gamma Function, 5.7 Euler’s beta function, 5.8 Taylor Series, 5.9 Fourier Series as Power Series, 5.10 Binomial Series, 5.11 Logarithmic Series, 5.12 Dirichlet Series and the Zeta Function, 5.13 Bernoulli Numbers and Polynomials, 5.14 Asymptotic series, 5.15 Fractional and complex derivatives, 5.16 Some electrostatic problems, 5.17 Infinite products, 6.1 Analytic functions, 6.2 Cauchy-Riemann conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula

14 Sep Wednesday

6.1 Analytic Functions, 6.2 Cauchy-Riemann Conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula

15 Sep Thursday

6.5 Harmonic Functions, 6.6 Taylor Series for Analytic Functions, 6.7 Cauchy’s Inequality, 6.8 Liouville’s Theorem, 6.9 Fundamental Theorem of Algebra, 6.10 Laurent Series, 6.11 Singularities, 6.12 Analytic Continuation, 6.13 Calculus of residues

20 Sep Tuesday

6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots

21 Sep Wednesday

6.17 Conformal mapping, 6.18 Cauchy’s principal value, 6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, 6.22 Method of steepest descent

22 Sep Thursday

Example of using Mathematica and Matlab to find inverse Laplace transforms, 6.22 Method of Steepest Descent, 7.1 Ordinary linear differential equations, 7.2 Linear partial differential equations, 7.3 Separable partial differential equations, 7.4 First-order differential equations, 7.5 Separable first-order differential equations

4 Oct Tuesday

Matlab tricks. Linear first-order ordinary differential equations, Small oscillations, Systems of ordinary differential equations, Exact higher-order differential equations, Constant-coefficient differential equations, Second-order ordinary differential equations, Frobenius’s solutions of second-order differential equations.

5 Oct Wednesday

Solved examples from chapters 6 & 7, and applications of Matlab.

6 Oct Thursday

Matlab examples of a chaotic autonomous system of ode's. Frobenius’s solutions of second-order differential equations, Fuch’s theorem, Even and odd differential operators, Wronski’s determinant, Second Solutions, Why not three solutions? Boundary conditions.

For now, you may ignore what lies below this line:

Link to information about FERPA and classroom recordings & media.

Link to actual release form.

You can use Adobe Reader or Acrobat to sign this form.

If all students registered for 466 & 468 sign and email this release form to me,

then I will post the videos of the classes on YouTube which would make them more accessible.

UNM Administrative Mandate on Required Vaccinations:

All students, staff, and instructors are required by the
UNM
Administrative Mandate on Required Vaccinations
to be fully vaccinated for COVID-19 as soon as possible, but no later
than September 30, 2022, and must provide proof of vaccination or of a
UNM validated limited exemption or exemption no later than September
30, 2021 to the
UNM vaccination
verification site. Students seeking medical exemption from the
vaccination policy must submit a request to the UNM verification site
for review by the UNM Accessibility Resource Center. Students seeking
religious exemption from the vaccination policy must submit a request
for reasonable accommodation to the
UNM verification
site
for review by the
Compliance, Ethics, and Equal Opportunity Office.
For further information on the requirement and on limited exemptions
and exemptions, see the
UNM Administrative Mandate on Required Vaccinations.

UNM Requirement on Masking in Indoor Spaces

All students, staff, and instructors are required to wear face masks in indoor classes, labs, studios and meetings on UNM campuses, see masking requirement. Qualified music students must follow appropriate specific mask policies issued by the Chair of the Department of Music and the Dean of the College of Fine Arts. Vaccinated and unvaccinated instructors teaching in classrooms must wear a mask when entering and leaving the classroom and when moving around the room. When vaccinated instructors are able to maintain at least six feet of distance, they may choose to remove their mask for the purpose of increased communication during instruction. Instructors who are not vaccinated (because of an approved medical or religious exemption), or who are not vaccinated yet, must wear their masks at all times. Students who do not wear a mask indoors on UNM campuses can expect to be asked to leave the classroom and to be dropped from a class if failure to wear a mask occurs more than once in that class. With the exception of the limited cases described above, students and employees who do not wear a mask in classrooms and other indoor public spaces on UNM campuses are subject to disciplinary actions.

Communication on change in modality: The President and Provost of UNM may direct that classes move to remote delivery at any time to preserve the health and safety of the students, instructor and community. Please check your email regularly for updates about our class and please check https://bringbackthepack.unm.edu regularly for general UNM updates about COVID-19 and the health of our community.

UNM Requirement on Masking in Indoor Spaces

All students, staff, and instructors are required to wear face masks in indoor classes, labs, studios and meetings on UNM campuses, see masking requirement. Qualified music students must follow appropriate specific mask policies issued by the Chair of the Department of Music and the Dean of the College of Fine Arts. Vaccinated and unvaccinated instructors teaching in classrooms must wear a mask when entering and leaving the classroom and when moving around the room. When vaccinated instructors are able to maintain at least six feet of distance, they may choose to remove their mask for the purpose of increased communication during instruction. Instructors who are not vaccinated (because of an approved medical or religious exemption), or who are not vaccinated yet, must wear their masks at all times. Students who do not wear a mask indoors on UNM campuses can expect to be asked to leave the classroom and to be dropped from a class if failure to wear a mask occurs more than once in that class. With the exception of the limited cases described above, students and employees who do not wear a mask in classrooms and other indoor public spaces on UNM campuses are subject to disciplinary actions.

Communication on change in modality: The President and Provost of UNM may direct that classes move to remote delivery at any time to preserve the health and safety of the students, instructor and community. Please check your email regularly for updates about our class and please check https://bringbackthepack.unm.edu regularly for general UNM updates about COVID-19 and the health of our community.

A list of errata for the second edition of

I am correcting typos and errors as I become aware of them and have added two new chapters,

chapter 23 on SI, natural, Planck, and Hartree units and chapter 24 on quantum mechanics.

Here are Chapters 1 to 4.

Here is latest version of Chapter 12 on special relativity.

Here is latest version of Chapter 13 on general relativity.

Here is latest version of Chapter 15 on probability and statistics.

Here is Chapter 23 on SI and natural units.

Here is Chapter 24 on quantum mechanics.

Three pages about black holes.

Here is an ongoing scroll of class notes from my iPad.

Here is a scroll started on 16 September.

Here is a scroll started on 5 October.

Here is a scroll started on 21 October.

Here is a scroll started on 16 November.

Here is the SAGE website which lets you download SAGE for free

and also provides links to videos and tutorials on how to use SAGE.

Here is a three-page description of how to get and use LaTeX.

Here is LaTeX tutorial.

The grader for the course is Mr. Andrew Forbes.

Please use email aforbes@unm.edu to send him your homework.

Homework 1 due Sunday 5 September:

Do problems 1.3, 1.8, 1.12, 1.15, 1.18, and 1.20.

Homework 2 due Sunday 12 September:

Do problems 1.25, 1.28, 1.32, 1.36, and 1.40, but in problem 1.32 use the matrix A = [2 3 4; 4 5 6].

Typos in problem 1.40 are corrected in the errata for the second edition of

Homework 3 due Sunday 19 September:

Do problems 2.1, 2.2, 2.3, 2.6, and 3.4.

Homework 4 due Sunday 26 September:

Do problems 3.5, 3.6, 3.16, 3.21, and 3.23.

Homework 5 due Sunday 3 October:

Do problems 4.5, 4.9, 4.12, 4.15, and 4.17.

Homework 6 due Sunday 10 October:

Do problems 5.1, 5.2, 5.5, 5.12, and 5.15.

Homework 7 due Sunday 17 October:

Do problems 6.5, 6.6, 6.7, 6.10, and 6.16.

Homework 8 due Sunday 24 October:

Do problems 6.20, 6.25, 6.28, 6.31, and 6.33.

Homework 9 due Sunday 31 October:

Do problems 7.9, 7.11, 7.12, 7.13, and 7.14.

Homework 10 due Sunday 7 November:

Do problems 7.19, 7.28, 7.35, 7.37, and 7.39.

I am told that one student is using a set of solutions rather than doing the homework himself. Therefore the problems I will assign in all future homeworks will be new problems not in my book. It will take me some time to invent and solve these new problems. Here is the homework assignment for next Sunday:

Homework 11 due Sunday 14 November:

Problem 1: Suppose a voltage $V(t) = V \cos(\omega t)$ is applied to a resistor of $R$ $ (\Omega)$ in series with an inductor of $L$ (H). If the current through the circuit at time $t=0$ is zero, what is the current at time $t$? You may use the equation $ L \dot I + R I ={} V(t)$.

Problem 2: If a voltage $V$ is applied to a series consisting of a resistor of $R$ $ (\Omega)$, an inductor of $L$ (H), and a capacitor of $C$ (F), then the current $I(t)$ obeys the equation $L C \ddot I + RC \dot I + I ={} C \dot V$. If the voltage $V$ across the series vanishes, and the current $I(t)$ and its first derivative at time $t=0$ are $I_0$ and 0, what is the current at time $t$? Complex notation makes this problem easier.

Problem 3: In a region of empty space where the pressure $ p $ and the chemical potentials $ \mu_j $ vanish, the change in the internal energy $ U = {} c^2 M $ of a black hole of mass $ M $ is $ dU = c^2 dM = {} T dS $ where $ S $ is its entropy, and $ T = \hbar \, c^3/(8\pi \, k \, G \, M)$ is its temperature. Find the entropy of a black hole of mass $M$.

Problem 4: If $\dot y(t) ={} \exp(-y(t))$ and $y(0)=1$, what is $y(t)$? You may use Matlab.

Problem 5: Find the function $u(x)$ that satisfies $$ - x^2u''(x) + 3xu'(x) + 2u(x) = 0 $$ with $u(0) = 0$ and $ u(1) = 1$. You may use Matlab.

Examples of the use of Matlab to solve ordinary differential equations.

Homework 12 due Sunday 21 November:

Problem 1:

Use section 8.1 to find the integral equation on the interval $[0,1]$ for the spherical Bessel functions $j_\ell(x)$ which obey Bessel's equation (10.67) $$ {} - \left( x^2 j'_\ell(x) \right)' - x^2 j_\ell(x) = {} - \ell(\ell+1) j_\ell(x). $$ How to convert an ode to an integral equation.

Problem 2: Find a solution of the ode $$ (1-x^2) \, f'' - 2x \, f' = {} - 6 \, f $$ that is nonsingular for $-1 \le x \le 1$. (Typo fixed: $6f \to {} - 6 f$.)

Problem 3: Evaluate numerically the counterclockwise contour integral around the origin $ z = 0 $ \begin{equation} I_n = \frac{1}{2^n \, 2\pi i} \oint \frac{(z^2 - 1 )^n} {z^{n+1}} \, dz \end{equation} for all integers $n$.

Problem 4: Find a solution to the equation \begin{equation} {} - \triangle V = {} - \frac{1}{\rho} \, \left [ \left ( \rho \, V_{,\rho} \right)_{,\rho} + \frac{ 1 }{ \rho } \, V_{,\phi\phi} + \rho \, V_{,zz} \right] = a^2 \, V \end{equation} for a problem in which the region $\rho \ge 0$ and all $z$ are physical and also a solution for a problem in which the region $\rho \ge 0$ and only all $z > 0$ are physical and also a solution for a problem in which the region $\rho \ge 0$ and only all $z < 0$. Altogether three solutions.

Problem 5: Find a solution to the equation \begin{equation} {} \triangle V = {} \frac{1}{\rho} \, \left [ \left ( \rho \, V_{,\rho} \right)_{,\rho} + \frac{ 1 }{ \rho } \, V_{,\phi\phi} + \rho \, V_{,zz} \right] = a^2 \, V \end{equation} for a problem in which the $\rho \ge 0$ and all $z$ are in the physical region

and also a solution for a problem in which the $\rho \ge 0$ and only all $z > 0$ are in the physical region and also a solution for a problem in which the $\rho \ge 0$ and only all $z < 0$ are in the physical region. Altogether three solutions.

Homework 13 due Sunday 28 November:

Problem 1: (a) Do the positive real numbers form a group under addition? If not, why not?

(b) Do the positive real numbers form a group under multiplication? If not, why not?

(c) Do the $4 \times 4$ matrices that are Lorentz transformations (12.8) form a group? If not, why not?

Problem 2: What would be the lifetime of a negative pion $\pi^-$ moving in a cyclotron at speed $v/c = 0.9$? (Here the nu is meant to be a v.) At rest its lifetime is $\tau ={} 2.6033 \times 10^{-8}$ s.

Latest version of chapter on special relativity.

Problem 3: Two photons collide head on and make a proton and an antiproton. What was the minimum energy that each photon had to have? The mass of a proton is 938.272081 MeV.

Problem 4: (a) Are the events noon on Earth and noon on the Sun timelike, spacelike, or null?

(b) A proton of energy 6.5 TeV in the LHC passes the same point $\boldsymbol x$ at times $t_1$ and $t_2$. Are the events, $(t_1, \boldsymbol x)$ and $(t_2, \boldsymbol x)$ timelike, spacelike, or null?

Problem 5: What is the 4-momentum $p^i$ at time $t$ of a particle of mass $m$ whose position at time $t$ is $\boldsymbol x \exp(t/t_0)$?

Latest version of chapter on general relativity.

Homework 14 due Sunday 5 December:

Problem 1: A hypothetical kinetic era in which kinetic energy is the main form of energy is sometimes called

During such a kinetic era, the energy density $\rho$ is the pressure divided by two factors of the speed of light, $\rho = p/c^2$, so $w=1$. How does the energy density $\rho$ vary with the scale factor? Hint: Use conservation of energy (13.279-13.284).

Problem 2: Suppose a particle of 4-momentum $p$ and mass $m > 0$ with $p^2 = {} \boldsymbol p^2 - (p^0)^2 = {} - c^2 m^2$ absorbs a photon.

Can the final state be the same particle with the same mass $m$ but with a different 4-momentum $q$ such that $q^2 = {} \boldsymbol q^2 - (q^0)^2 = {} - c^2 m^2$? If not, why not?

Problem 3: Under the general coordinate transformation $$ x'^i ={} L \, e^{x^i/L} $$ in which $L$ is a fixed length, (a) what is $A'^j(x')$ if $A^k(x)$ is a contravariant vector field, and

(b) what is the partial derivative $$ \partial_{k'} = \frac{\partial}{\partial x'^k}? $$

Problem 4: In the movie

on a planet near a black hole and after returning to Earth, finds his daughter who was 10 when he left, to be 80 years of age. If the planet was 1 au ~ $ 1.5 \times 10^{11} $ m from the black hole, what was the mass of the black hole? Hint: The proper time $d\tau$ measured by a clock at rest at a distance $r$ (in Schwarzschild's coordinates) from a mass $M$ is \begin{equation} d\tau ={} \sqrt{ 1 - \frac{2GM}{c^2 r} } \, dt \end{equation} in which $dt$ is the time measured by a clock at rest far from the mass $M$.

Problem 5: Assume that the first period after inflation is an era of kination in which the parameter $k=0$. Find how the scale factor $a$ depends upon the time $t$. Hint: The section Nonlinear Differential Equations in Cosmology (7.46 in PM) and problem 1 above may be useful.

Homework 15 due Sunday 12 December:

Problem 1: By filling in the steps skipped between equations (13.214) and (13.217), show that the simpler action principle \begin{equation} 0 ={} \delta \int_{\tau_1}^{\tau_2} g_{i \ell} (x) \, u^i \, u^\ell \, d\tau ={} \delta \int_{\tau_1}^{\tau_2} g_{i \ell} (x) \, \frac{dx^i}{d \tau} \, \frac{dx^\ell}{d \tau} \, d\tau . \label {simpler action principle} \end{equation} yields the geodesic equation $$ 0 ={} \frac{d^2 x^r}{d\tau^2} + \Gamma^r_{\phantom{r} i\ell } \, \frac{d x^i}{d\tau} \, \frac{d x^\ell }{d\tau} . $$ The equations numbers are those of the latest version of chapter on general relativity.

Problem 2: Find the value of the integral \begin{equation} I = \int_{-\infty}^\infty \frac{1}{(x - i) (x - 2 i) (x + 3i)} \, dx. \end{equation}

Problem 3: Find the evolution of the scale factor $a(t)$ for a universe composed of a single substance with a constant but arbitrary real value of $w = p/(c^2 \rho)$. Assume $k=0$ and $a(0) = 0$ and $a(t_0) \equiv a_0 = 1$ in which $t_0$ is the present time. Hint: Reread the section Nonlinear Differential Equations in Cosmology.

Problem 4: Suppose we ask three likely voters if they will vote for Michael Heinrich, and two say "Yes." What is the probability that he will be re-elected? Hint: Imitate example 15.4.

Problem 5: The first two zeros of $J_0(x)$ are $z_{0,1} = 2.4048$, and $z_{0,2} = 5.5201$. The first two zeros of $J_1(x)$ are $z_{1,1} = 3.8317$, and $z_{1,2} = 7.0156$. The first two zeros of $J_2(x)$ are $z_{2,1} = 5.1356$, and $z_{2,2} = 8.4172$. The frequencies of the TM modes of a cylindrical cavity of height $h$ and radius $r$ are $$ \omega_{n,m,\ell} = c \sqrt{z_{n,m}^2/r^2 + \pi^2 \ell^2 /h^2}. $$ Assume $h=2r$. The lowest frequencies are for $\ell = 1$. Find the value of the radius $r$ for which the energy gap between the ground state (of one photon) and the first excited state (of one photon) is 10 eV. (We can think of this can of photons as a photonic atom.) Hint: Use example 10.4 and set $ \hbar \Delta \omega = 10$ eV.

I will be updating what follows, which is from 466 for 2020.

Homework 1 due Sunday 30 August:

Do problems 1.1, 1.5, 1.11, 1.15, 1.19, & 1.20.

Homework 2 due Sunday 6 September:

Do problems 1.25, 1.28, 1.32, 1.34, & 1.35.

Homework 3 due Sunday 13 September:

Do problems 1.40, 2.1, 2.2, 3.2, 3.16.

Homework 4 due Sunday 20 September:

Do problems 3.17, 3.21, 3.25.

Homework 5 due Sunday 27 September:

Do problems 4.5, 4.6, 4.9, 4.15, and 4.16.

Homework 6 due Sunday 4 October:

Do problems 5.2, 5.4, 5.12, 5.19, and 6.1.

Homework 7 due Sunday 11 October:

Do problems 6.8, 6.15, 6.18, 6.20, and 6.33.

Homework 8 due Sunday 18 October:

Do problems 6.21, 6.24, 6.25, 6.30, and 6.34.

Homework 9 due Sunday 25 October:

Do problems 6.35, 7.2, 7.3, 7.4, 7.5, and 7.8.

Homework 10 due Sunday 1 November:

Do problems 7.7, 7.9, 7.10, 7.15, and 7.19.

Homework 11 due Sunday 8 November:

Do problems 7.10, 7.29, 7.30, 7.32, and 7.33.

Homework 12 due Sunday 15 November:

Do problems 7.25, 7.34, 7.35, 8.1, and 8.2.

Homework 13 due Sunday 22 November:

Do problems 8.3; 9.2 but only for $n = 0, 1$, and 2; 9.5; 9.12; and 9.13.

Homework 14 due Sunday 29 November:

Do problems 9.14, 10.3, and 10. 13.

Homework 15 due Sunday 6 December:

Do problems 9.15, 10.15, 10.18, 10.23, and 10.26.

Homework 16 due Thursday 10 December:

3.4, 4.10, 5.22, and 6.5.

Some examples for some of the chapters

and class notes for chapters 3 and 4; chapters 1, 2, and 4; chapter 6; chapters 6 and 7;

468 class of 14 October; 466 class of 15 October; 466/468 iPad notes of 28 October 2020 ;

466/468 iPad notes of 4 November 2020; iPad notes starting on 24 Nov 2020; and iPad notes starting on 2 Dec 2020.

Each date is the starting date of its set of notes.

My pedagogical paper on the CMB and my paper on Lorentz bosons and dark matter.

Here is Chapter 2.

Here is Chapter 3.

A list of errata for the second edition of

Corrected and improved version of Example 4.13 (Lifetime of a Fluorophore).

Here's a link to a folder containing videos of the problem sessions of physics 468.

Videos of lectures of physics 466:

18 August 2020.

Linear Algebra: 1.1 Numbers, 1.2 Arrays, 1.3 Matrices, 1.4 Vectors,1.5 Linear operators, 1.6 Inner products. Examples of outer products, eigenvectors and eigenvalues. Illustrated examples of the use of Matlab.

20 August 2020.

1.8 Linear independence and completeness 1.9 Dimension of a vector space 1.10 Orthonormal vectors 1.11 Outer products 1.12 Dirac notation 1.13 Adjoints of operators 1.14 Self-adjoint or hermitian linear operators 1.15 Real, symmetric linear operators 1.16 Unitary operators 1.17 Hilbert spaces

25 August 2020.

1.18 Antiunitary, antilinear operators 1.19 Symmetry in quantum mechanics 1.20 Determinants 1.21 Jacobians 1.22 Systems of linear equations 1.23 Linear least squares 1.24 Lagrange multipliers 1.25 Eigenvectors and eigenvalues 1.26 Eigenvectors of a square matrix Also, the LU decomposition and examples of how to use Matlab to find eigenvalues, eigenvectors, determinants, and LU decompositions of matrices.

27 August 2020.

1.26 Eigenvectors and eigenvalues 1.27 Eigenvectors of a square matrix 1.28 A matrix obeys its characteristic equation 1.29 Functions of matrices 1.30 Hermitian matrices 1.31 Normal matrices 1.32 Compatible normal matrices 1.33 Singular-value decompositions 1.34 Moore-Penrose pseudoinverses 1.35 Tensor products and entanglement

1 September 2020.

2 Vector calculus: 2.1 Derivatives and partial derivatives 2.2 Gradient 2.3 Divergence 2.4 Laplacian 2.5 Curl 3 Fourier series 3.1 Fourier series 3.2 The interval 3.3 Where to put the 2\(\pi\)’s 3.4 Real Fourier series for real functions

3 September 2020.

3 Fourier series: 3.1 Fourier series, 3.2 The interval, 3.3 Where to Put the 2pi’s, 3.4 Real Fourier series for real functions, 3.5 Stretched intervals, 3.6 Fourier series of functions of several variables, 3.7 Integration and differentiation of Fourier series, 3.8 How Fourier series converge, 3.9 Measure and Lebesgue integration (barely mentioned), 3.10 Quantum-mechanical examples, 3.11 Dirac’s delta function

3 September 2020.

3.10 Quantum-mechanical examples, 3.11 Dirac’s delta function, 3.12 Harmonic Oscillators, 3.13 Nonrelativistic Strings, 3.14 Periodic Boundary Conditions. Also, the Helmholtz decomposition and a generalization of Fourier series.

10 September 2020.

4 Fourier and Laplace transforms, 4.1 Fourier transforms, 4.2 Fourier transforms of real functions, 4.3 Dirac, Parseval, and Poisson, and some remarks about Lebesgue integration and generalized Fourier series.

15 September 2020.

4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables, 4.6 Convolutions, 4.7 Fourier transform of a convolution, 4.8 Fourier transforms and Green’s functions, 4.9 Laplace transforms, 4.10 Inversion of Laplace transforms, 4.11 Volterra’s Convolution, 4.12 Derivatives and integrals of Laplace transforms, 4.13 Laplace transforms and differential equations, and 4.14 Applications to Differential Equations.

17 September 2020.

5 Infinite series: 5.1 Convergence, 5.2 Tests of convergence, 5.3 Convergent series of functions, 5.4 Power series, 5.5 Factorials and the gamma function, 5.6 Euler’s beta function, 5.7 Taylor series, 5.8 Fourier series as power series, 5.9 Binomial series, 5.10 Logarithmic series, 5.11 Dirichlet series and the zeta function, 5.12 Bernoulli numbers and polynomials, 5.13 Asymptotic series, and 5.16 Infinite products.

22 September 2020.

6 Complex-variable theory: 6.1 Analytic functions, 6.2 Cauchy-Riemann conditions, 6.3 Cauchy’s integral theorem, and 6.4 Cauchy’s integral formula.

24 September 2020.

6.4 Cauchy’s integral formula, 6.5 Harmonic functions, 6.6 Taylor series for analytic functions, 6.7 Cauchy’s inequality, 6.8 Liouville’s theorem, 6.9 Fundamental theorem of algebra, 6.10 Laurent series, and 6.11 Singularities.

29 September 2020.

6.10 Laurent series, 6.11 Singularities, 6.12 Analytic continuation, 6.13 Calculus of residues, and 6.14 Ghost contours

1 October 2020.

6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots, 6.17 Conformal mapping, and 6.18 Cauchy’s principal value.

6 October 2020.

6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, and 6.22 Method of steepest descent; 7 Differential equations, 7.1 Ordinary linear differential equations, and 7.2 Linear partial differential equations.

8 October 2020 .

7.3 Separable partial differential equations, 7.4 First-order differential equations, 7.5 Separable first-order differential equations, 7.6 Hidden separability, 7.7 Exact first-order differential equations, and 7.8 Meaning of exactness.

13 October 2020 .

7.7 Exact first-order differential equations, 7.8 Meaning of exactness, 7.9 Integrating factors, 7.10 Homogeneous functions, 7.11 Virial theorem, 7.12 Legendre’s transform, 7.13 Principle of stationary action in mechanics, 7.14 Symmetries and conserved quantities in mechanics. Also some remarks about the relationship of exact differentials to the Cauchy-Riemann equations.

15 October 2020 .

7.15 Homogeneous first-order ordinary differential equations, 7.16 Linear first-order ordinary differential equations, 7.17 Small oscillations, 7.18 Systems of ordinary differential equations, 7.19 Exact higher-order differential equations, 7.20 Constant-coefficient differential equations, 7.21 Singular points of second-order ordinary differential equations, and 7.22 Frobenius’s series solutions.

21 October 2020 .

7.21 Singular points of second-order ordinary differential equations, 7.22 Frobenius’s series solutions, 7.23 Fuch’s theorem, 7.24 Even and odd differential operators, 7.25 Wronski’s determinant, 7.26 Second Solutions, 7.27 Why not three solutions?, 7.28 Boundary conditions, 7.29 A variational problem, 7.30 Self-adjoint differential operators, and 7.31 Self-adjoint differential systems.

22 October 2020 .

7.32 Making operators formally self adjoint, 7.33 Wronskians of Self-Adjoint Operators, 7.34 First-order self-adjoint differential operators, 7.35 A constrained variational problem, 7.36 Eigenfunctions and eigenvalues of self-adjoint systems, and 7.37 Unboundedness of eigenvalues.

28 October 2020 .

7.37 Unboundedness of eigenvalues, 7.38 Completeness of eigenfunctions, 7.39 Inequalities of Bessel and Schwarz, and 7.40 Green’s functions.

29 October 2020 .

7.32 Making operators formally self adjoint, 7.36 Eigenfunctions and eigenvalues of self-adjoint systems, 7.37 Unboundedness of eigenvalues, 7.38 Completeness of eigenfunctions, 7.40 Green’s functions, 7.41 Eigenfunctions and Green’s functions, 7.43 Principle of stationary action in field theory, 7.44 Symmetries and conserved quantities in field theory, 7.45 Nonlinear differential equations, 7.46 Nonlinear differential equations in cosmology, and 7.47 Nonlinear differential equations in particle physics.

4 November 2020 .

The hydrogen atom. 8 Integral equations: 8.1 Differential equations as integral equations, 8.2 Fredholm integral equations, 8.3 Volterra integral equations, 8.4 Implications of linearity, and 8.5 Numerical solutions.

5 November 2020 .

7.44 Symmetries and conserved quantities in field theory, 8.1 Differential equations as integral equations, 8.2 Fredholm integral equations, 8.3 Volterra integral equations, 8.4 Implications of linearity, 8.5 Numerical solutions, 8.6 Integral transformations, 9 Legendre polynomials and spherical harmonics, 9.1 Legendre’s polynomials, 9.2 The Rodrigues formula, 9.3 Generating function for Legendre polynomials, 9.4 Legendre’s differential equation, 9.5 Recurrence relations, 9.6 Special values of Legendre polynomials, 9.7 Schlaefli’s integral, and 9.8 Orthogonal polynomials.

10 November 2020 .

Wave mechanics of composite objects, 9.8 Orthogonal polynomials, 9.9 Azimuthally symmetric laplacians, 9.10 Laplace’s equation in two dimensions, 9.11 Helmholtz’s equation in spherical coordinates, 9.12 Associated Legendre polynomials, 9.13 Spherical harmonics, and 9.14 Cosmic microwave background radiation.

11 November 2020 .

Why the eigenvalues of Bessel's equation are positive, How to derive the formulas for the evolution of the scale factor for a universe of radiation, and How one uses an integral transformation to solve Bessel's equation.

12 November 2020 .

10 Bessel Functions, 10.1 Bessel Functions of the First Kind, Modified Bessel functions, Surface of a drum, Cylindrical wave guides, and Cylindrical cavities.

17 November 2020 .

10.2 Bessel Functions of the Second Kind, 10.3 Bessel Functions of the Third Kind, 10.4 Spherical Bessel Functions of the First Kind, and 10.5 Spherical Bessel Functions of the Second Kind.

19 November 2020 .

Review of singular-value decomposition, Fourier series and transforms, gamma function, Cauchy's theorem and formula, Legendre polynomials, and Bessel functions. Introduction to Monte Carlo methods.

24 November 2020 .

Binomial distribution, Poisson's distribution, Gauss's distribution, Monte Carlo methods, and Special relativity.

25 November 2020 .

Spherical Bessel functions and Helmholtz's equation, Particle in a small sphere, Review of FLRW cosmologies, Scalars, Covariant and contravariant vectors, Basis vectors and the metric tensor.

1 December 2020 . Review of general coordinate invariance, basis vectors, metric tensor, covariant derivatives, and Christoffel symbols. Solving a first-order linear differential equation, Spherical Bessel functions at small and large arguments, electrostatic potential inside and outside a hollow sphere, and spherical-harmonic expansion of a plane wave.

2 December 2020 . A little more about tangent vectors, maximally symmetric spaces, the geodesic equation, Einstein's equations, Schwarzschild's solution, black holes, and FLRW cosmologies.

3 December 2020 . Examples of cylindrical wave guides, Fourier series, Fourier-Laplace transforms, radii of convergence, and contour integration. Some remarks about probability and Bayes's theorem. The low-base-rate problem, the three-door problem, a tiny poll, quantum mechanics, and transitivity. Average values and their variances.

10 December 2020 . Inner products, the Cauchy-Schwarz inequality, and the triangle inequality. Singular-value decomposition. Density operators. Hamilton systems, integrability, autonomous systems, attractors, chaos, fractals, and strange attractors. Example of artificial intelligence. Path integrals and the connection between quantum and classical physics. Aharonov-Bohm effect.

The grader for the course is Mr. Evgeni Zlatanov.

Please use email to return your homework to him at zlatanov@unm.edu.

All homework problems are stated in the book

The best way to do your homework is to use latex to make pdf files and to use email to send the grader your pdf files.

TeXShop works well on Apple computers. You can get TeXShop here pages.uoregon.edu/koch/texshop/.

TeXstudio works well on Windows computers. You can get TeXstudio here www.texstudio.org.

Both use TeX Live which you can get here www.tug.org/texlive/acquire-netinstall.html.

TENTATIVE SYLLABUS

Here is what I plan to cover in this course:

Linear algebra: 2 weeks

Vector calculus 0.5

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 are the first ten chapters of the book.

Welcome to physics 466 for 2019.

Class meets in room 184 of the physics building at 1919 Lomas NE at 5:30 pm on Tuesdays and Thursdays.

The problem session for the course, physics 468, will meet in room 5 from 5 to 5:50 and not in room 1131 as originally scheduled.

We will be using the second edition of a textbook called

You can get it now from Amazon and eBooks.

The book is now available in the UNM bookstore.

Here is Chapter 1.

Here is Chapter 2.

Here is Chapter 3.

A list of errata in the second edition of

Corrected and improved version of Example 4.13 (Lifetime of a Fluorophore).

SYLLABUS

Here is what I plan to cover in this course:

Linear algebra: 2 weeks

Vector calculus 0.5

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 are the first ten chapters of the book.

All homework problems are stated in the book

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

You can send me e-mail.

Homework 1 due Friday 30 August:

Do problems 1-3, 5-7, & 9-14 of chapter 1.

Homework 2 due Friday 6 September:

Do problems 15-22, 25, 27-31 of chapter 1.

Homework 3 due Tuesday 17 September:

Do problems 1.32-1.36 and 1.40 and 2.2-2.6 of chapter 2.

Homework 4 due Friday 27 September:

Do problems 3.1, 3.2, 3.4-3.12, and for extra credit 3.16-3.21 of chapter 3.

Homework 5 due Friday 4 October:

Do problems 4.1-4.9 of chapter 4.

Homework 6 due Tuesday 15 October:

Do problems 4.10-4.18 of chapter 4.

Homework 7 due Friday 25 October:

Do problems 5.1-5.5 of chapter 5 and problems 6.1, 6.3, 6.5, and 6.6 of chapter 6.

Homework 8 due Friday 1 November:

Do problems 6.7, 6.8, 6.11, 6.13, 6.15, 6.16, 6.20, and 6.24 of chapter 6.

Homework 8 due Friday 8 November:

Do problems 6.28, 6.30, 6.33, 6.34, 6.35, and 6.38 of chapter 6, and 7.2 and 7.9 of chapter 7.

Homework 9 due Friday 15 November:

Do problems 7.10 -- 7.15, 7.17, and 7.19.

Homework 10 due Monday 25 November:

Do problems 7.25 -- 7.27, 7.29--7.30, and 7.32--7.34.

Homework 11 due Monday 9 December:

Do problems 9.2 (but only for \(n=0, 1,\) and 2), 9.8, 9.14, 9.17, 9.18, 10.1, 10.3, 10.13, 10.15, 10.18.

The final exam is on Thursday 12 December from 5:30 to 7:30 in our regular classroom 1160.

Videos of lectures:

20 August

Linear algebra: Sections 1.1 Numbers, 1.2 Arrays, 1.3 Matrices, 1.4 Vectors, 1.5 Linear operators, 1.6 Inner products, 1.7 Cauchy–Schwarz inequalities, and 1.8 Linear independence and completeness.

22 August

1.9 Dimension of a vector space, 1.10 Orthonormal vectors, 1.11 Outer products, 1.12 Dirac notation, 1.13 Adjoints of operators, 1.14 Self-adjoint or hermitian linear operators, 1.15 Real, symmetric linear operators.

27 August

1.16 Unitary operators, 1.17 Hilbert spaces, 1.18 Antiunitary and antilinear operators, 1.19 Symmetry in quantum mechanics, 1.20 Determinants, 1.21 Jacobians, 1.22 Systems of linear equations, 1.23 Linear least squares, and 1.24 Lagrange multipliers.

29 August

1.24 Lagrange multipliers, 1.25 Eigenvectors and eigenvalues, 1.26 Eigenvectors of a square matrix, 1.27 A matrix obeys its characteristic equation, 1.28 Functions of matrices, and. 1.29 Hermitian matrices.

3 September

1.30 Normal matrices, 1.31 Compatible normal matrices, 1.32 Singular-value decompositions, 1.33 Moore-Penrose pseudoinverses, 1.34 Tensor products and entanglement, 1.35 Density operators, 1.36 Schmidt decomposition, 1.37 Correlation functions, 1.38 Rank of a matrix, and 1.39 Software.

5 September

2.1 Derivatives and partial derivatives, 2.2 Gradient, 2.3 Divergence, 2.4 Laplacian, and 2.5 Curl

10 September

3.1 Fourier series, 3.2 The interval, 3.3 Where to put the 2pi’s, 3.4 Real Fourier series for real functions, 3.5 Stretched intervals, 3.6 Fourier series of functions of several variables, 3.7 Integration and differentiation of Fourier series, and 3.8 How Fourier series converge.

12 September

3.9 Measure and Lebesgue integration, 3.10 Quantum-mechanical examples, 3.11 Dirac’s delta function, 3.12 Harmonic oscillators, 3.13 Nonrelativistic strings, and 3.14 Periodic boundary conditions.

19 September

4.1 Fourier transforms, 4.2 Fourier transforms of real functions, 4.3 Dirac, Parseval, and Poisson, 4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables, 4.6 Convolutions, 4.7 Fourier transform of a convolution, 4.8 Fourier transforms and Green’s functions, 4.9 Laplace transforms, 4.10 Derivatives and integrals of Laplace transforms, 4.11 Laplace transforms and differential equations, and 4.12 Inversion of Laplace transforms.

24 September

Review of Sections 4.1-4.12 and discussion of Section 4.13 Application to differential equations.

26 September

5.1 Convergence, 5.2 Tests of convergence, 5.3 Convergent series of functions, 5.4 Power series, and 5.5 Factorials and the gamma function.

1 October

5.5 Factorials and the gamma function, 5.6 Euler’s beta function, 5.7 Taylor series, 5.8 Fourier series as power series, 5.9 Binomial series, 5.10 Logarithmic series, 5.11 Dirichlet series and the zeta function, 5.12 Bernoulli numbers and polynomials, 5.13 Asymptotic series, 5.14 Fractional and complex derivatives, 5.15 Some electrostatic problems, 5.16 Infinite products, 6.1 Analytic functions, 6.2 Cauchy-Riemann conditions, and 6.3 Cauchy’s integral theorem.

3 October

6.1 Analytic functions, 6.2 Cauchy-Riemann conditions, 6.3 Cauchy’s integral theorem, 6.4 Cauchy’s integral formula, and 6.5 Harmonic functions.

8 October

6.5 Harmonic functions, 6.6 Taylor series for analytic functions, 6.7 Cauchy’s inequality, 6.8 Liouville’s theorem, 6.9 Fundamental theorem of algebra, 6.10 Laurent series, 6.11 Singularities, 6.12 Analytic continuation, and 6.13 Calculus of residues.

15 October

6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots, 6.17 Conformal mapping, 6.18 Cauchy’s principal value, and 6.19 Dispersion relations.

22 October

Sections 6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, and 6.22 Method of steepest descent.

24 October

7.1 Ordinary linear differential equations, 7.2 Linear partial differential equations, 7.3 Separable partial differential equations, 7.4 First-order differential equations, and 7.5 Separable first-order differential equations.

29 October

7.6 Hidden separability, 7.7 Exact first-order differential equations, 7.8 Meaning of exactness, 7.9 Integrating factors, 7.10 Homogeneous functions, 7.11 Virial theorem, and 7.12 Legendre’s transform.

31 October

7.12 Legendre’s transform, 7.13 Principle of stationary action in mechanics, 7.14 Symmetries and conserved quantities in mechanics, 7.15 Homogeneous first-order ordinary differential equations, 7.16 Linear first-order ordinary differential equations, 7.17 Small oscillations, 7.18 Systems of ordinary differential equations, 7.19 Exact higher-order differential equations, and 7.20 Constant-coefficient equations.

5 November

7.21 Singular points of second-order ordinary differential equations, 7.22 Frobenius’s series solutions, 7.23 Fuch’s theorem, 7.24 Even and odd differential operators, 7.25 Wronski’s determinant, 7.26 Second solutions, 7.27 Why not three solutions?, 7.28 Boundary conditions, 7.29 A variational problem, and 7.30 Self-adjoint differential operators.

7 November

Introduction to Maxima by Logan Cordonnier, 7.31 Self-adjoint differential systems, 7.32 Making operators formally self adjoint, 7.33 Wronskians of self-adjoint operators, 7.34 First-order self-adjoint differential operators, and 7.35 A constrained variational problem.

12 November

7.35 A constrained variational problem, 7.36 Eigenfunctions and eigenvalues of self-adjoint systems, 7.37 Unboundedness of eigenvalues, 7.38 Completeness of eigenfunctions, 7.39 Inequalities of Bessel and Schwarz, 7.40 Green’s functions, 7.41 Eigenfunctions and Green’s functions, and 7.42 Green’s functions in one dimension.

14 November

7.43 Principle of stationary action in field theory, 7.44 Symmetries and conserved quantities in field theory, 7.45 Nonlinear differential equations, 7.46 Nonlinear differential equations in cosmology, and 7.47 Nonlinear differential equations in particle physics.

19 November

8.1 Differential equations as integral equations, 8.2 Fredholm integral equations, 8.3 Volterra integral equations, 8.4 Implications of linearity, 8.5 Numerical solutions, 8.6 Integral transformations, and 9.1 Legendre’s polynomials, 9.2 The Rodrigues formula, 9.3 Generating function for Legendre polynomials, 9.4 Legendre’s differential equation, 9.5 Recurrence relations, 9.6 Special values of Legendre polynomials, 9.7 Schlaefli’s integral, and 9.8 Orthogonal polynomials.

21 November

9.8 Orthogonal polynomials, 9.9 Azimuthally symmetric laplacians, 9.10 Laplace’s equation in two dimensions, 9.11 Helmholtz’s equation in spherical coordinates, 9.12 Associated Legendre polynomials, 9.13 Spherical harmonics, 9.14 Cosmic microwave background radiation, and 10.1 Cylindrical Bessel functions of the first kind.

26 November

Bessel functions of the first kind, Bessel functions of the second kind, Bessel functions of the third kind, Spherical Bessel functions of the first kind, and Spherical Bessel functions of the second kind.

3 December

Solutions to some of the exercises of the chapter on Legendre polynomials and spherical harmonics, and a quick introduction to general relativity.

5 December

Solutions to some of the exercises on Bessel functions.

All students of physics should read at least the first section of the essay The Trouble with Quantum Mechanics by Steven Weinberg before they graduate.