--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.13.6 kernelspec: display_name: Python 3 (phys-571) language: python name: phys-571 --- ```{code-cell} :tags: [hide-cell] %pylab inline import numpy as np, matplotlib.pyplot as plt ``` (sec:ClassLog)= # Class Log - 2025 These are notes about what we did in class in the Fall 2025 offering of the course. ## Mon 1 Dec 2025 * Multi-pole Expansion. ## Wed 12 Nov 2025 * Discussed the catenary problem (hanging chain). See [Classical Mechanics Notes on Functional Derivatives][] for complete details. * Also discussed the Legendre-Fenchel transform for a saturating equation of state (see [Classical Mechanics Notes on Droplets][] for details). Morals: 1. Look for analogies with Classical Mechanics where you have intuition and rich tools. 2. Look for symmetries. In this case, the functional was lacking a variable, leading directly to a conservation law. [Classical Mechanics Notes on Functional Derivatives]: [Classical Mechanics Notes on Droplets]: ## Mon 10 Nov 2025 * Calculus of Variations. Introduced the catenary problem (hanging chain). See [Classical Mechanics Notes on Functional Derivatives][] for complete details. ## Fri 7 Nov * Central limit theorem. (See {ref}`sec:RG-RandomWalks` for details.) * Monte-Carlo integration. Importance sampling. Metropolis algorithm. ## Wed 5 Nov * Bayesian analysis. See {ref}`sec:BayesianAnalysis` for details. ## Mon 3 Mov 2025 * Probability. * Monte Hall problem. * Bayes theorem. * Meaning of $\chi^2$ fitting as Bayesian analysis with a flat (uninformed) prior and gaussian errors. ## Fri 24 Oct 2025 * Euler-Maclaurin Formula. * Mobius transformation. * Group theory review (Bosons and Fermions). ## Wed 22 Oct 2025 * Perturbation theory and Asymptotic Series. * Mittag-Lefler Theorem and $\zeta(2k)$. ## Mon 20 Oct 2025 * Orthogonal Polynomials. * Stone-Weierstrass Theorem. * Gaussian Quadrature. * Bernoulli Numbers. ## Fri 17 Oct 2025 * Saddle-point integration and the gamma function $z! = \int_0^{\infty} t^ze^{-t}\d{t}$. ## Wed 15 Oct 2025 * \begin{gather*} \int_0^{\infty} \frac{\sin x}{x}\d{x} \end{gather*} ## Mon 13 Oct 2025 Mostly covered stuff in {ref}`sec:ComplexAnalysis`. * $\sin(1+\I)$. * $27^{1/3}$ (all values). * Fundamental theorem of algebra. * Radius of convergence of $G(x) = 1/(1-x^2)$ and $H(x) = 1/(1+x^2)$. * Analytic continuation of $1/(1-x)$ to $x<-1$. * {ref}`sec:ComplexDerivatives`. * {ref}`sec:ContourIntegrals`. * {ref}`sec:Residues`. ## Fri 10 Oct 2025 * Review of Green's Functions. * Complex Analysis. * Complex numbers. \begin{gather*} z = x + \I y = r e^{\I\theta}.\\ \frac{1}{1-\I} = \frac{1}{1-\I}\frac{1+\I}{1+\I} = \frac{1+\I}{1-(\I)^2} = \frac{1}{2} + \frac{1}{2}\I.\\ e^{z} = \sum_{n=0}^{\infty} \frac{z^n}{n!}, \qquad \ln(z) = \Log(z) + 2\pi \I n.\\ e^{x+\I y} = e^{x}e^{\I y} = e^{x}(\cos y + \I \sin y).\\ z^w = (e^{\ln(z)})^w = e^{w\ln(z)} = e^{w\Log(z) + 2\pi \I n w}. \end{gather*} * Multifunctions (Riemann sheets). * $\ln z$ vs $\Log(z)$. * $e^z$ vs $(2.718\dots)^z$. * Clausen's Paradox: $(e^a)^b = e^{ab} \implies 1 = e^{-2\pi}$ for $a=2\pi \I$ and $b=\I$. Resolved by defining $(e^a)^b = e^{b \ln e^a} = e^{b (a + 2\pi \I n)}$. Then $(e^{2\pi \I})^{\I} = e^{\I(2\pi \I + 2\pi \I n)}$, which gives $1^\I = e^{0}$ for $n=-1$ as we normally expect. ## Wed 8 Oct 2025 * Green's Functions: For alternative approaches and discussions, see [The Pendulum][]. (Please read.) [The Pendulum]: ## Mon 6 Oct 2025 * Midterm Exam 1 ## Wed 1 Oct 2025 * Review of updated notes on {ref}`sec:Frobenius`. * General discussion about PDEs: see {ref}`sec:PDEs` for details. ## Mon 29 Sept 2025 * Discussion of linear algebra, change of bases, and Dirac notation. ## Fri 26 Sept 2025 * Discussion of {ref}`sec:SturmLiouville`. ## Wed 24 Sept 2025 * Discussion of {ref}`sec:ODEs`. ## Mon 22 Sept 2025 * Introduction of {ref}`sec:ODEs`. ## Fri 19 Sept 2025 * Discussion of {ref}`sec:Tensors`. ## Wed 17 Sept 2025 * Discussion of {ref}`sec:Tensors`. ## Mon 15 Sept 2025 * Introduction to {ref}`sec:Tensors`. ## Fri 12 Sept 2025 * Helmholtz Decomposition. * Question: What does the following matrix do? \begin{gather*} \mat{M} = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}. \end{gather*} This corresponds to the coordinate transformation \begin{gather*} x' = x+y, \qquad y' = x \end{gather*} Find the eigenvalues and eigenvectors as quickly as possible. Note that $\mat{M}$ is hermitian. * Matrix factorization. * $UDV^\dagger$: Singular Value Decomposition (SVD). * Example: Entanglement (Schmidt decomposition) Did not get to the following: * Hermitian vs self-adjoint. Use finite difference operator as example. * $L_2$ (see {cite}`Hassani:2013` chapter 7, Thms. 7.2.1-7.2.3 * Riesz-Fischer: $\mathcal{L}^2_{w}(a,b)$ is complete. * All complete inner product spaces with countable bases are isomorphic to $\mathcal{L}^2_{w}(a,b)$. * Stone-Weierstrass: $\bigl\{x^n \mid n \in \{0, 1, 2, \dots\}\bigr\}$ forms a basis for $\mathcal{L}^2_{w}(a,b)$. * Pauli matrices. * Start on Tensors / Differential Geometry. * Metric. Curvilinear coordinates. Jacobian. ## Wed 10 Sept 2025 * Angular momentum. * Jacobian. * Differential forms. All vector derivative identities come from \begin{gather*} \int_{V} \d{f} = \int_{\partial V} f\\ \int_{L}f'(x)\d{x} = \int_{\partial L} f(x) = f(L_1) - f(L_0),\\ \int_{V}\vect{\nabla}\cdot \vect{F}\d^{3}x = \int_{\partial V} \vect{F}\cdot\d^2{\vect{A}},\\ \int_{A}(\vect{\nabla}\times\vect{F})\cdot\d^{2}\vect{A} = \int_{\partial A} \vect{F}\cdot\d{\vect{l}}. \end{gather*} * Operators representing conserved quantities generate the corresponding symmetry transformations: \begin{gather*} e^{\tau\op{H}/\I\hbar}\psi(t) = \psi(t + \tau),\\ e^{\vect{\lambda}\cdot\op{\vect{p}}/\I\hbar}\psi(\vect{x}) = \psi(\vect{x} - \vect{\lambda}),\\ e^{\vect{\theta}\cdot\op{\vect{L}}/\I\hbar}\psi(\vect{x}) = \psi(\mat{R}_\vect{\theta}^{-1}\vect{x}), \qquad \mat{R}_{\vect{\theta}} = e^{\mat{\vect{\theta}\times}}. \end{gather*} ## Mon 8 Sept 2025 * Discussion of vectors as curves, and some of the meaning of things like div, grad, and curl. (Incomplete) * Types of linear transforms: Scaling, Rotation, Shears. * Hermitian matrices have a complete orthonormal basis. * Completeness \begin{gather*} \mat{1} = \sum_{n}\ket{n}\bra{n} = \int \ket{x}\bra{x}\d{x} = \int \ket{k}\bra{k}\frac{\d{k}}{2\pi}, \qquad \braket{x|k} = e^{\I k x}, \end{gather*} *(using my normalizations)*. * Show that this gives the Fourier transform: \begin{gather*} \psi_k = \braket{k|\psi} = \braket{k|\mat{1}|\psi} = \int\braket{k|x}\braket{x|\psi}\d{x} = \int e^{-\I k x}\psi(x)\d{x} \end{gather*} * Jordan Normal Form * SVD * Question: What does the following matrix do? \begin{gather*} \mat{M} = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}. \end{gather*} This corresponds to the coordinate transformation \begin{gather*} x' = x+y, \qquad y' = x \end{gather*} ## Fri 5 Sept 2025 * Rotation matrices. * Linear approximations. * Quick overview of Groups and Lie Algebras. ## Wed 3 Sept 2025 * Levi-Civita symbol. * Vectors and matrices, especially projection operators. ## Fri 29 Aug 2025 ```{code-cell} :tags: [margin, hide-input] # Numerical check import numpy as np, matplotlib.pyplot as plt rng = np.random.default_rng(seed=2) Ns = 100000 x = rng.normal(0.5, 0.25, size=Ns) a = 1.0 z = x - a fig, ax = plt.subplots(1, 1, figsize=(4,3)) kw = dict(bins=int(np.sqrt(Ns)/2), density=True, histtype="step") ax.hist(x, linestyle='-', label="$p(x)$", **kw) ax.hist(z, linestyle='--', label="$p_z(z)$", **kw) ax.hist(x**2, linestyle=':', label="$p_{x^2}(x^2)$", **kw) ax.legend() ax.set(xlabel="$x$, $z$", ylabel="PDF"); ``` :::{margin} Here $p(x)$ is a gaussian with mean $\mu=0.5$ and standard deviation $\sigma_x = 0.25$ (solid), along with the PDFs of $z = x-1.0$ (dashed) is shift and the PDF of $x^2$ (dotted) is quite different. The latter is a $\chi^2$ distribution. ::: Problem for the day (from Wed): **Given a random variable $x$ with PDF $p(x)$, what is the PDF $p_z(z)$ for $z = f(x)$?** For example, if $f(x) = x-a$, then you should find $p_z(z) = p(z+a)$. **plan** * Gram-Schmidt: * Mentioned that functions are vectors. \begin{gather*} \braket{f|g} = \int \d{x} f^*(x) g(x) \textcolor{green}{w(x)}. \end{gather*} Unifies fourier, orthogonal polynomials, many special functions, etc. * Some power series: \begin{gather*} \ln(1-x) = x + \frac{x^2}{2} + \cdots + \frac{x^{n}}{n} + \cdots.\\ \end{gather*} * Dirac delta function. Distribution. * Finite differences. ## Wed 27 Aug 2025 Discussion of adding errors in quadrature: I.e., exactly what it means to say that if $x = \bar{x} \pm \sigma_x$ and $y = \bar{y} \pm \sigma_y$, then \begin{gather*} z = x + y = \overbrace{\bar{x} + \bar{y}}^{\bar{z}} \pm \overbrace{\sqrt{\sigma_x^2 + \sigma_y^2}}^{\sigma_z}. \end{gather*} Demonstrate that if $x$ and $y$ are independent random variables with PDFs $p_x(x)$ and $p_y(y)$, then $z = x+y$ has PDF \begin{gather*} p_z = p_x * p_y, \qquad p_z(z) = \iint\!\d{x}\d{y}\; \delta\bigl(z - (x+y)\bigr)p_x(x)p_y(y) = \int\!\d{y}\;p_x(z-y)p_y(y), \end{gather*} I.e., $p_z$ is the [convolution][] of $p_x$ and $p_y$. Mentioned (no derivation) the following logic: 1. $x = \bar{x} \pm \sigma_x$ means $x \sim \mathcal{N}_{\mu=\bar{x}, \sigma=\sigma_x}$ is normally distributed with mean $\bar{x}$ and standard deviation $\sigma_x^2$. 2. The PDF $p_x(x)$ is thus a gaussian with variance $\sigma_x^2$. 3. The Fourier transform of this is also a gaussian with variance $1/\sigma_x^2$. (This is a manifestation of the uncertainty principal.) 4. The [convolution theorem][] tells us that the Fourier transform of $p_z = p_x*p_y$ is just the project of the Fourier transforms: \begin{gather*} \mathcal{F}(p_z) = \mathcal{F}(p_x)\mathcal{F}(p_y). \end{gather*} 5. The PDF $p_z(z)$ is thus a gaussian with covariance $\sigma_z^2 = \sigma_x^2 + \sigma_y^2$: i.e. the **covariances add**. This follows directly from the fact that \begin{gather*} \mathcal{F}(p_{x}) \propto e^{-k^2 \sigma_{x}^2/2}, \qquad \mathcal{F}(p_{z}) = \mathcal{F}(p_{x})\mathcal{F}(p_{y}) \propto e^{-k^2 \sigma_{x}^2/2} e^{-k^2 \sigma_{x}^2/2} = e^{-k^2 (\sigma_{x}^2 + \sigma_y^2)/2}. \end{gather*} Mentioned that even for non-Gaussian variables, the covariances add $\sigma_z^2 = \sigma_x^2 + \sigma_y^2$, but that the shape of the distribution might change. Mentioned that this also holds for linear transformations $z = ax + by +c$. Since every "nice" function is approximately linear if you zoom in far enough, once you have small enough errors you can use the quadrature rule. But don't forget the limitations: 1. The errors are small enough that the function is linear over the range of errors. 2. The errors are independent. 3. The errors are gaussian. *(Not strictly needed, but implicit in the notation $\mu\pm \sigma$. * Mentioned the [Weierstrass M-test][] and [Abel's test][], discussed Cauchy convergence, and how limits might not exist, e.g., the real numbers (with cardinality $\aleph_1$: i.e. uncountable) can be formed as the collection of convergent Cauchy sequences of rational numbers (which have cardinality $\aleph_0$: i.e. countable). [convolution]: [convolution theorem]: [Weierstrass M-test]: [Abel's test]: ## Mon 25 Aug 2025 * Inner product spaces: Some confusion. (E.g. Notion of angle in complex spaces?) * Discussion of matrices and vectors. $(\mat{A}\mat{B})^\dagger = \mat{B}^\dagger\mat{A}^\dagger$. * Index notation to make sure: \begin{gather*} [(\mat{A}\mat{B})^\dagger]_{ik} = A_{kj}^*B_{ji}^* = B_{ji}^*A_{kj}^* = [\mat{B}^\dagger]_{ij}[\mat{A}^\dagger]_{jk} = [\mat{B}^\dagger\mat{A}^\dagger]_{ik}. \end{gather*} ## Fri 22 Aug 2025 * Feynman's differentiation trick. * Series, convergence. * Review of better method of series convergence: integral test + generalized ratio test. See notes in {ref}`sec:MathematicalPreliminaries`. * Alternating series that are not absolutely convergent can sum to anything. * Mentioned that functions are vectors. \begin{gather*} \braket{f|g} = \int \d{x} f^*(x) g(x) \textcolor{green}{w(x)}. \end{gather*} Unifies fourier, orthogonal polynomials, many special functions, etc. ## Wed 20 Aug 2025 * Introductions. * Technical problem: Are the following limits convergent or divergent? \begin{gather*} u_{n} = \frac{1}{n^s (\ln n)^t}, \qquad \lim_{N\rightarrow\infty}\int^{N} u_n\d{n}, \qquad \lim_{n\rightarrow\infty} \frac{u{n+1}}{u_{n}}. \end{gather*} * Quick review of linear algebra. * Need review of Inner Product Spaces, SVD, etc. * Did the matrix exponential: \begin{gather*} e^{\mat{M}} = \lim_{N\rightarrow \infty} \sum_{n=0}^{N} \frac{\mat{M}^{n}}{n!} \end{gather*} * Discussed how to learn: how I learn. * Quick review of Syllabus, reading assignment.