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Linear Algebra
Following is the detailed chapter-wise outline for the Linear Algebra book focused on physical applications.
Page Contents
Vector Differentiation
Vector differentiation is the mathematical process of determining how vector quantities change with respect to a scalar variable, most commonly time or space, providing a precise language to describe motion, flow, and field variation in physics.
Hooke’s Law
Hooke’s Law is one of the foundational principles of classical mechanics and elasticity theory, describing the linear relationship between the force applied to an elastic body and the resulting deformation, provided the deformation remains within the elastic limit of the material. Formulated in the 17th century by the English scientist Robert Hooke, the law captures the essential behavior of springs, wires, rods, and a wide class of solid materials when subjected to small stresses. In its simplest and most widely used form, Hooke’s Law states that the restoring force developed in an elastic system is directly proportional to the displacement from its equilibrium position and acts in the opposite direction.
Scalar and Vector Products
Scalar (dot) and vector (cross) products are fundamental binary operations between vectors that yield, respectively, a scalar measuring directional alignment and a vector representing oriented area and rotational tendency, forming the mathematical backbone of geometry, mechanics, and field theory.
Vector Algebra
Vector Algebra is the mathematical framework that deals with quantities possessing both magnitude and direction and provides systematic rules for their representation, manipulation, and combination, forming the backbone of physical descriptions of space, motion, and fields.
Structure of Formal Salutation
The first and most crucial stage of any academic lecture or formal speech is the formal salutation. This is the point at which the speaker brings together the stage, the audience, and the occasion into a single coherent frame. A well-crafted salutation not only establishes the seriousness and tone of the speech but also creates a sense of discipline, attentiveness, and expectation among the listeners.
औपचारिक अभिवादन
किसी भी अकादमिक व्याख्यान या औपचारिक भाषण का पहला और सबसे महत्वपूर्ण चरण औपचारिक अभिवादन होता है। यह वह बिंदु है जहाँ वक्ता मंच, श्रोता और अवसर—तीनों को एक सूत्र में बाँधता है। सही ढंग से रचा गया अभिवादन न केवल भाषण की गंभीरता स्थापित करता है, बल्कि श्रोताओं में अनुशासन और अपेक्षा की भावना भी उत्पन्न करता है।
सेवानिवृत्ति भाषण: 2026
निम्नलिखित 31 जनवरी 2026 को इतिहास-कक्ष-1, एसकेएमयू दुमका में आयोजित डॉ. हस्मत अली और डॉ. संजय कुमार सिन्हा के सेवानिवृत्ति समारोह के दौरान प्रस्तुत किए गए सेवानिवृत्ति भाषणों के अंश हैं:
Talk on AI
In 1950, Alan Turing published a seminal paper titled “Computing Machinery and Intelligence,” in which he asked the provocative question: “Can machines think?” This question laid the foundation for the field of artificial intelligence (AI) and has since sparked decades of research and debate.
Coherent States
There are several distinct definitions and constructions of coherent states in the literature, each with its own mathematical formulation, physical interpretation, and domain of applicability. Below are some of the most prominent types of coherent states, along with their definitions, mathematical formulations, descriptions, applications, and foundational references.
Upgrading an Old Jekyll Project to Jekyll 4: Full Error Diagnosis and Fix
This project was originally built using Jekyll 3.x and worked correctly for several years. Later, the local system Ruby environment was upgraded and Jekyll 4 was installed. This guide documents the complete process of diagnosing and resolving incompatibilities during the migration.
Quantum Information: NonG Measure
The non-Gaussianity (nonG) of a continuous-variable (CV) quantum state $ \rho $ is defined as the quantum relative entropy distance between $ \rho $ and a reference Gaussian state $ \rho_G $ that has the same first moments and the same covariance matrix as $ \rho $:
Quantum Information: Weigner Non-Gaussianity Measure
A celebrated phase-space description of nonclassicality in single-mode quantum oscillators is based on the presence of negative regions of the Wigner function.
Since the Wigner function is a normalized but not positive-definite quasi-probability distribution, its negativity has no classical counterpart.
Franck Condon Principle
The Franck–Condon principle is one of the most fundamental concepts in molecular spectroscopy, explaining why vibrational structures appear in electronic spectra of molecules and why certain transitions are more intense than others. When a molecule undergoes an electronic transition—whether by absorption or emission of radiation—the change in the electronic state occurs on a timescale much faster than nuclear motion. Electrons are extremely light compared to nuclei; therefore, their transitions happen almost instantaneously relative to the vibrational and rotational movement of the nuclei. As a consequence of this difference in timescales, the nuclei can be considered “frozen” during the electronic transition. This approximation is the core of the Franck–Condon principle and leads to a vertical transition between potential energy curves on a Born–Oppenheimer energy diagram.
Scattering: Partial Wave Analysis
Partial wave analysis is a fundamental method in quantum scattering theory used to analyze the interaction of a particle with a localized potential by exploiting the rotational symmetry of the problem. When a quantum particle of definite momentum is incident on a scattering center, its wavefunction far from the interaction region can be expressed as a superposition of an incoming plane wave and an outgoing spherical wave.
Chokes and Transformers
Chokes and transformers are fundamental electromagnetic components widely used in electrical and electronic systems, particularly in power supplies, communication circuits, and signal-conditioning networks. Both devices operate on the principles of electromagnetic induction and magnetic flux linkage, yet they serve distinct functional roles within circuits. A choke is essentially an inductor designed primarily to impede alternating current (AC) while allowing direct current (DC) to pass with minimal resistance. In contrast, a transformer is a static electrical device that transfers electrical energy between two or more circuits through mutual induction, usually with the purpose of changing voltage or current levels, or providing electrical isolation.
RF and AF Oscillators
Oscillators are fundamental electronic circuits capable of generating periodic waveforms without the need for an external input signal. They operate by converting direct current (DC) power into alternating current (AC) signals through the use of active devices such as transistors, operational amplifiers, or vacuum tubes, in conjunction with passive components like resistors, capacitors, and inductors. Depending on the frequency range of the generated signal, oscillators are broadly classified into Audio Frequency (AF) oscillators and Radio Frequency (RF) oscillators. AF oscillators typically generate signals in the range of approximately 20 Hz to 20 kHz, which corresponds to the human audible spectrum. These oscillators are widely used in audio signal generators, public address systems, audio testing equipment, and musical instruments. RF oscillators, on the other hand, operate at much higher frequencies, typically from hundreds of kilohertz to several gigahertz, and form the backbone of radio communication systems, including transmitters, receivers, radar, television broadcasting, and wireless communication technologies.
Capacitors
A capacitor is a fundamental passive electronic component used to store electrical energy in the form of an electric field. It consists essentially of two conducting surfaces (plates) separated by an insulating medium known as a dielectric. When a potential difference is applied across the plates, equal and opposite charges accumulate on them, giving rise to an electric field within the dielectric. The ability of a capacitor to store charge per unit potential difference is quantified by its capacitance, measured in farads (F). Capacitors are indispensable in both DC and AC circuits and play a crucial role in signal processing, power conditioning, filtering, timing, coupling, decoupling, and energy storage.
Breadboard Usage
A breadboard is one of the most fundamental and widely used tools in experimental electronics and applied physics laboratories, especially at the undergraduate and postgraduate levels. It serves as a temporary construction platform for prototyping, testing, and analyzing electronic circuits without the need for soldering. The term “breadboard” originates historically from early experimental setups where wooden boards (sometimes literally breadboards) were used to mount electronic components. Modern breadboards, however, are standardized plastic boards with internal metallic spring contacts arranged in a highly structured manner.