Posts
Quantum Angular Momentum
Angular momentum in quantum mechanics is not merely the quantization of $\mathbf{L}=\mathbf{r}\times\mathbf{p}$. That formula describes one specific realization (orbital angular momentum). The deeper unifying idea is that angular momentum is the generator of rotations acting on quantum states in Hilbert space, and orbital angular momentum, spin, and total angular momentum are different representations of the same rotational structure.
TikZ Diagram Prompts
For TikZ/PGF physics diagrams, the most reliable workflow is to use AI to generate editable TikZ code rather than a final image. This preserves mathematical correctness, enables systematic refinement, and supports consistent styling across an entire textbook with export-ready output (PDF/SVG/PNG).
Asymptote Diagram Prompts
For Asymptote physics diagrams, the most reliable workflow is to use AI to generate editable Asymptote code that you can refine, reuse, and export as publication-quality vector graphics (PDF/SVG/PNG). Asymptote is especially valuable when diagrams require 3D geometry, controlled perspective, surfaces, coordinate systems, spatial trajectories, or technical linework beyond typical 2D textbook schematics.
Mechanics Concept Map
Classical mechanics can be read as a continuous shift from a force-first description to a structure-first description. Each stage keeps the same physical predictions but changes the language so that constraints, symmetries, and conserved quantities become simpler to express. The progression moves from forces and accelerations to variational principles, then to phase-space geometry, and finally to generators and Poisson-bracket algebra.
Free Science Journals
A good undergraduate science reader should be clear, conceptually rich, reliable, and freely accessible in digital format. The following curated list brings together high-quality journals and science magazines that are especially useful for students, teachers, and early researchers.
Conservation Theorems and Symmetry Properties
In Hamiltonian mechanics, canonical transformations are important because they preserve the form of Hamilton’s equations. If the old canonical variables $(q_i,p_i)$ are replaced by new variables $(Q_i,P_i)$, the transformation is canonical only when the new variables also satisfy Hamilton’s equations in the same structural form. This allows us to change variables without changing the basic geometry of mechanics.
Poisson Bracket & Theorems
The Poisson bracket is one of the central tools of Hamiltonian mechanics. It gives a compact way to describe time evolution, canonical transformations, symmetries, and conservation laws. Once the Hamiltonian formulation is written in terms of canonical variables $(q_i,p_i)$, the Poisson bracket becomes the natural mathematical operation that connects phase-space functions with physical motion.
Canonical Maps
Hamilton–Jacobi theory naturally extends to canonical transformations because the action function $S$ itself behaves like a generating function. This extension clarifies why Hamiltonian mechanics can be simplified by changing phase-space variables and how continuous symmetry transformations arise from infinitesimal generators.
Hamilton–Jacobi Equation
The Hamilton–Jacobi formulation rewrites dynamics in terms of a single function $S$, called Hamilton’s principal function. Instead of solving the equations of motion directly, one solves a first-order partial differential equation for $S$ and then extracts the trajectory from it. This method is powerful because it connects variational mechanics, Hamiltonian mechanics, and the integration of motion in one framework.
Legendre Transform
As we have seen in the previous lectures, the Lagrangian and Hamiltonian formulations of mechanics are related by a mathematical operation called the Legendre transformation. This transformation is not just a technical tool; it is a fundamental concept that appears in various areas of physics, including thermodynamics and classical mechanics. In this lecture, we will explore the Legendre transformation in depth, understand its motivation, and see how it is applied in different physical contexts.
Hamilton’s Equations of Motion
This lecture contains principle of least action, Hamilton’s equations of motion, and solved examples.
Particle Physics-II
In particle reactions and decays, certain quantities remain unchanged because they arise from fundamental symmetries of nature. These conservation laws act as selection rules: if a proposed process violates a conserved quantity for the interaction responsible (strong, electromagnetic, or weak), then the process is forbidden or strongly suppressed. A clear way to learn particle physics is to first master which quantities are conserved in which interactions, and then practice applying them to specific decays and reactions.
Particle Physics-I
- In modern particle physics, forces are explained as interactions via exchange of particles.
- These particles are called field particles, exchange particles, or gauge bosons.
- Interaction between two particles occurs through continuous emission and absorption of field particles.
- Force is not action at a distance but mediated by particle exchange.
Calogero–Sinha Case
We study a two-degree-of-freedom model with a velocity-coupling term and an inverse-square interaction. The classical dynamics becomes transparent after a change of variables that separates a conserved “drift-like” quantity from an Ermakov–Pinney-type radial equation. Quantization in the same variables leads to a solvable singular-oscillator equation whose normalizability requires contour (Stokes-wedge) boundary conditions rather than the real axis.
GNU plot preamble
Maxima can pass raw gnuplot directives to the plotting backend through gnuplot_preamble, enabling fine control over plot aesthetics. This approach is especially useful for scientific figures where borders, ticks, grids, labels, legends, numeric formatting, and sampling density must be standardized.
Lagrangian Mechanics
Lagrangian mechanics reformulates dynamics in terms of generalized coordinates $q_i(t)$ and a single scalar function $L(q_i,\dot{q}_i,t)$, the Lagrangian. Rather than starting from forces, one derives the equations of motion by demanding that the physical trajectory makes a functional called the action stationary among all nearby paths with the same endpoints.
Resonance Scattering
In nuclear scattering, an incident particle approaches a target nucleus, interacts through the nuclear (and possibly Coulomb) forces, and then emerges in some outgoing channel. The experimentally measured quantity is the differential or total cross-section, which quantifies the probability of scattering per target nucleus. For spherically symmetric (central) interactions, the most systematic and physically transparent description is the partial-wave method, where the scattering amplitude is resolved into contributions labelled by orbital angular momentum $l$.
Resonance reactions lie between the extremes of direct reactions and compound nucleus reactions. They involve discrete, quasibound nuclear states in the energy spectrum.
Compound-Nucleus Reactions
When an incident particle approaches a target nucleus with impact parameter smaller than the nuclear radius, it can interact strongly with individual nucleons. After the initial encounter, the incident particle and recoiling nucleon undergo successive collisions inside the nucleus, progressively redistributing energy among many degrees of freedom. With small probability, a nucleon (or light cluster) acquires sufficient energy to escape, in close analogy with evaporation from a heated liquid.
Maxima Tables and Loops
Maxima supports disciplined, readable computational workflows by combining list generation, matrix constructors, and loop-based iteration with formatted printing. In practice, tabular output is produced with print(a, b, c) or printf(true, “format”, args), while symbolic objects can be displayed in Maxima’s two-dimensional mathematical form via the ~m format directive.
Maxima Algebra Simplification
Maxima provides a compact set of commands for rewriting algebraic expressions into cleaner equivalent forms, including rational simplification, polynomial factorization/expansion, fraction manipulation, and common trigonometric or exponential rewrites.