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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.
Classical Orthogonal Polynomials
Classical orthogonal polynomials occupy a central place in mathematical physics, approximation theory, and special functions. In Maxima, these families are available through direct symbolic commands, allowing compact access to their standard forms as well as to associated and generalized variants.
Maxima Scientific Plots
GNU PREAMBLE provides a powerful way to customize the appearance of Maxima plots by injecting raw gnuplot commands. This allows for precise control over borders, ticks, legends, grids, and sampling density, enabling the creation of clean, publication-quality figures without relying on terminal-specific directives.
Maxima Matrix Essentials
Maxima supports standard matrix construction and linear-algebra operations needed for symbolic and exact computations, including linear transformations, coupled systems, and matrix methods in quantum mechanics. The commands below form a minimal toolkit for routine matrix algebra, determinants and inverses, linear systems, and spectral analysis.
Calculus with Maxima
Maxima provides a compact, symbolic workflow for standard calculus operations—differentiation, integration, limits, summation/product manipulation, series expansion, algebraic equation solving, and differential equation solving—supporting both routine computation and physics-oriented analytical work.
PT-Symmetric Radial Oscillator
Exact Solution on Shifted Contour
Starting from the radial Schrödinger equation on the complex-shifted contour $r=x-i\varepsilon,; x\in(-\infty,\infty),; \varepsilon>0$,
Python: Course Contents
Python is a high-level, interpreted programming language designed for clarity and efficiency. It allows a programmer to express ideas with minimal syntax while maintaining strong computational capability. A Python program executes sequentially, and each instruction operates on data to produce meaningful outcomes. Its simplicity makes it suitable for beginners, while its extensive libraries support advanced scientific and analytical tasks.
Symmetry and Algebra in Physics
A defining insight of modern theoretical physics is that the fundamental laws of nature are governed not merely by differential equations, but by symmetry principles. These symmetries are not handled directly at the level of transformations alone; instead, they are encoded in algebraic structures built from generators. The passage from geometric transformations to algebra is what allows physics to extract universal, coordinate-independent content. In this chapter, we explore this idea in depth through three major examples: rotations, translations, and supersymmetry.
Lie Superalgebras in Supersymmetry
Why Study Lie Superalgebras in Supersymmetry
Harmonic Oscillator and Lie Superalgebra
The harmonic oscillator provides the simplest setting where operator factorization leads naturally to supersymmetric structure. The Hamiltonian is written as