Category: Index
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Ehrenfest’s Theorem
Introduction Ehrenfest’s theorem is a fundamental principle in quantum mechanics that reveals a deep connection between quantum mechanics and classical mechanics. It essentially bridges the gap between the quantum and classical descriptions of physical systems. Basics of Ehrenfest’s Theorem Ehrenfest’s theorem states that the average value of an observable (such as position or momentum) in…
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Density Functional Theory (DFT)
Introduction Density Functional Theory (DFT) is a computational quantum mechanical modelling method used to investigate the electronic structure of many-body systems, especially atoms, molecules, and the condensed phases. The Basics of DFT The fundamental premise behind DFT is the Hohenberg-Kohn theorem, which states that the ground state properties of a many-electron system are uniquely determined…
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Dirac Matrices
Introduction The Dirac matrices, or gamma matrices, are a set of matrices with specific anticommutation relations that play a key role in the Dirac equation, a relativistic quantum mechanical wave equation discovered by British physicist Paul Dirac. Definition of Dirac Matrices The Dirac matrices are defined in four-dimensional Minkowski space and are generally denoted as…
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Biot–Savart Law
Introduction The Biot–Savart Law is a fundamental principle in electromagnetism that describes the magnetic field generated by an electric current. Named after physicists Jean-Baptiste Biot and Félix Savart, the law provides a mathematical description of the magnetic field in terms of the current source that produces it. Statement of the Biot–Savart Law The Biot–Savart Law…
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Gromov’s Theorem
Introduction Gromov’s Theorem, proposed by mathematician Mikhail Gromov, is a fundamental result in the field of geometric group theory, which studies the algebraic properties of groups through their actions on geometric spaces. There are several theorems attributed to Gromov, but in geometric group theory, the most renowned one is perhaps the Gromov’s Theorem on groups…
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Isotropy
Introduction Isotropy is a fundamental concept in physics that describes a system which has the same properties in all directions. The term is derived from the Greek ‘isos’, meaning equal, and ‘tropos’, meaning direction. Understanding isotropy aids in describing a variety of physical phenomena from the distribution of galaxies in the universe to the properties…
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Casimir-Polder Force
Introduction The Casimir-Polder force is a physical force arising from quantum fluctuations of the electromagnetic field in vacuum. It was first described by Dutch physicists Hendrik Casimir and Dirk Polder in 1948, who were investigating the interactions between molecules and a dielectric body. Quantum Electrodynamics and Vacuum Fluctuations Quantum electrodynamics (QED) tells us that even…
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Anisotropy
Introduction Anisotropy refers to the property of being directionally dependent. In other words, a material is said to be anisotropic if its physical or mechanical properties vary when measured along different directions. This is in contrast to isotropy, where properties do not change regardless of the direction in which they are measured. Mathematical Representation An…
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Aharonov–Bohm Effect
Introduction The Aharonov–Bohm effect is a quantum mechanical phenomenon that demonstrates how a charged particle is affected by electromagnetic fields, despite being in regions of apparently zero electromagnetic fields. This effect illustrates the fundamental role of potentials in quantum mechanics, in contrast to classical physics where fields play the dominant role. Aharonov–Bohm Electromagnetic Potential The…
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Acoustic Impedance
Introduction Acoustic impedance, denoted as , is a property of a medium which defines its opposition to the propagation of acoustic waves. It is a fundamental concept in fields such as acoustics, ultrasound, sonar, and other areas where sound propagation is crucial. Definition of Acoustic Impedance Acoustic impedance is a measure of the resistance of…