Class Log - 2025

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import numpy as np, matplotlib.pyplot as plt

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Populating the interactive namespace from numpy and matplotlib

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#

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.

Mon 10 Nov 2025#

Fri 7 Nov#

  • Central limit theorem. (See Renormalizing Random Walks for details.)

  • Monte-Carlo integration. Importance sampling. Metropolis algorithm.

Wed 5 Nov#

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 11. Complex Analysis.

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.)

Mon 6 Oct 2025#

  • Midterm Exam 1

Wed 1 Oct 2025#

Mon 29 Sept 2025#

  • Discussion of linear algebra, change of bases, and Dirac notation.

Fri 26 Sept 2025#

Wed 24 Sept 2025#

Mon 22 Sept 2025#

Fri 19 Sept 2025#

Wed 17 Sept 2025#

Mon 15 Sept 2025#

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 [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#

Hide code cell source

 
# 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");
_images/23c52a83458ca57c40c5088a1765f51ad264ac98695cf4f662b87a55c15615cc.png

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).

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 1. Mathematical Preliminaries.

  • 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.

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