Posts - Page 2 of 9
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
AI-Editable Physics Figures
For physics authors, the most reliable use of AI is not to generate final raster images, but to generate editable diagram code that can be corrected, refined, version-controlled, reused, and exported cleanly as PDF/SVG/PNG. This produces figures that remain mathematically faithful (angles, forces, labels, geometry) while keeping the author in full control of final presentation.
Nuclear Models
The atomic nucleus is a many-body quantum system consisting of protons and neutrons (nucleons) bound by the strong nuclear force. Because the exact interaction is complex, different nuclear models are used to explain various observed properties such as binding energy, stability, spin, and energy levels.
Superspace, Supergeometry, and Supersymmetry Transformations
The progression of theoretical physics has shown time and time again that advancements are generally made when the underlying mathematical structures utilized for describing nature have been expanded upon. Just as it become necessary to use a geometric concept of space-time to explain the change from classical to relativistic mechanics, as well as using linear Hilbert spaces to create quantum mechanics; in order to explain the concept of supersymmetry, we require an even more complex structure. This is superspace and it adds a set of anti-commuting variables to ordinary space-time coordinates which represent the fermionic degrees of freedom we have.
Supersymmetry and Lie Superalgebras
As modern physics has developed, it has shown that the expansion of the concept of symmetry has led to new insights into the nature of the fundamental forces of nature. While classical theories of physics, such as Newtonian mechanics, and quantum theories of physics, such as quantum mechanics, both rely on the transformation of objects through some sort of symmetry, there is still a major limitation to our understanding of matter and energy: bosons and fermions will always be treated as completely different particles, regardless of their behaviour under different conditions. When supersymmetry is introduced into the picture, it offers a novel way to relate the two types of particles by providing a new form of symmetry that relates these apparently dissimilar particles.
Lie Groups, Lie Algebras, and Quantum Structure
Recent developments in physics have highlighted the importance of continuous symmetry as a means to establish a connection between geometry and algebra and thus uncovering the underlying geometric structure that determines the physical laws. The concept of continuous symmetries originated with the invention of the theory of Lie groups and their associated Lie algebras by the mathematician Sophus Lie in the nineteenth century to describe smooth transformations like translations and rotations. In the twentieth century continuous symmetries became increasingly important in how they relate to physical theories, especially with the emergence of quantum mechanics and quantum field theory, where the fundamental physical laws and fundamental particles obey the same symmetry principles.
Foundations of Symmetry in Quantum Theory
Symmetry is fundamental to physics today, because it provides a common language bridging abstract mathematics with observable phenomena. In classical mechanics, symmetry provides the basis for the invariance of physical laws under rotation; and symmetry has become a central organizing principle - connecting various areas in physics - from very deep, structural constraints in quantum mechanics, through the use of quantum mechanics to define mathematically the structure of observed particles.
AI Tools
All AI tools mentioned in the 5-day AI workshop conducted at SBU Ranchi are listed below with their country of origin, purpose, open-source status, and alternatives.
JupyterLab
JupyterLab is a modern, web-based interactive development environment for Python and other languages. It combines notebooks, terminals, file management, and interactive tools into one unified workspace.
Python Setup
To do symbolic calculation safely and high-quality numerical work, the most practical stack is:
Frequently Used Terminal Commands
macOS is built on Unix, which means powerful terminal commands are available for file management, networking, system monitoring, and development tasks.
pyenv
Pyenv is an essential tool for Python developers, especially when working on multiple projects that require different Python versions. It allows you to easily switch between versions and maintain project-specific environments without affecting your system Python. Some key benefits of using pyenv include:
- Manage multiple Python versions (3.8, 3.9, 3.10, 3.11 etc.)
- Maintain project-specific environments
- Avoid breaking system Python
- Ensure team-wide version consistency