Physics 466

Physics 466, Fall 2021 Physical Mathematics

Welcome to physics 466 for 2021.

Due to the pandemic, all classes will be held online via Zoom at least until fall break.
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 will start at 5:30 on Tuesday and Thursday
and run to 6:45.
468 classes will start at 5:30 on Wednesday and run to 6:20.

Although the first real class will be on Tuesday 24 August,
I held a very brief trial class at 5:30 on Thursday, the 19th.

We will be using the second edition of a textbook called Physical Mathematics published in 2019 by Cambridge University Press.
New and used copies of the book are available in the UNM bookstore, but you can also get it from Amazon and eBooks.
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, 2021, 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.

A list of errata for the second edition of Physical Mathematics.
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 Physical Mathematics.
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 kination (unfortunately).
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 Interstellar, an astronaut spends — let us say a week —
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 Physical Mathematics.

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 Physical Mathematics.

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 Physical Mathematics published this summer by Cambridge University Press.
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 Physical Mathematics.
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 Physical Mathematics. Put homework in Evgeni Zlatanov's mailbox by 3:00 PM on its due date, usually a Friday. You can send him e-mail.
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.

There will be a midterm exam on the Thursday, 17 October, after fall break.
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.