Tag: Quantum
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Square-Integrable Functions
Introduction Square-integrable functions play a fundamental role in areas of physics such as quantum mechanics, signal processing, and optics. These are functions for which the integral of the square of the absolute value is finite. Definition A function defined over a measure space is said to be square-integrable if: This is commonly seen in the…
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Chapter 20: The Measurement Problem
20.1 Introduction to the Measurement Problem The measurement problem is a fundamental issue in quantum mechanics that arises from the apparent contradiction between the deterministic evolution of a quantum system’s wavefunction and the seemingly random outcomes observed during a measurement. The measurement problem raises questions about the interpretation of quantum mechanics and the role of…
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Chapter 19: The Harmonic Oscillator
19.1 Introduction to the Harmonic Oscillator The harmonic oscillator is a fundamental model in physics that describes the behavior of a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. In quantum mechanics, the harmonic oscillator is a key model used to understand a variety of physical systems,…
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Chapter 18: Particle in a Box
18.1 Introduction to Particle in a Box The particle in a box is a simplified model in quantum mechanics that helps to understand the behavior of a quantum particle confined in a one-dimensional, infinitely deep potential well. This model is important because it provides an elementary example of bound states, quantization of energy levels, and…
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Chapter 17: Introduction to Quantum Mechanics
17.1 Quantum Mechanics Overview Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It emerged in the early 20th century as a response to the limitations of classical mechanics and electromagnetism, which failed to describe the observed behavior of particles at the microscopic scale.…
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Chapter 16: The Uncertainty Principle
16.1 Introduction The uncertainty principle is a fundamental concept in quantum mechanics that places limits on the precision with which certain pairs of physical properties can be simultaneously known. It was first formulated by Werner Heisenberg in 1927 and is crucial for understanding the behavior of particles at the quantum level. This chapter will discuss…
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Tsirelson’s Bound
Introduction Tsirelson’s bound is a fundamental concept in the theory of quantum mechanics and quantum information theory. Named after the mathematician Boris Tsirelson, this bound provides a limit to the quantum mechanical correlation that can be achieved between two separated systems, playing a key role in distinguishing quantum theory from other possible physical theories. Mathematical…
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Spherical Harmonics
Introduction Spherical harmonics are mathematical functions that play a significant role in various fields, including quantum mechanics, electrodynamics, and computer graphics. They arise when solving Laplace’s equation in spherical coordinates, leading to a separation of variables solution. Definition and Mathematical Formulation of Spherical Harmonics Spherical harmonics can be defined in terms of the associated Legendre…
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Lamb Shift
Introduction The Lamb Shift, named after physicist Willis Lamb, refers to a slight difference in energy between two levels of the electron in a hydrogen atom, specifically the and states. This phenomenon was not predicted by the Dirac equation but was later explained by developments in quantum electrodynamics (QED). The Dirac Equation and Its Limitations…
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Klein Paradox
Introduction The Klein Paradox, proposed by Swedish physicist Oskar Klein, is a counterintuitive phenomenon in quantum mechanics and quantum field theory, particularly in the context of relativistic quantum mechanics described by the Dirac equation. The paradox refers to the penetration of high-energy particles through high potential barriers, contrary to the predictions of non-relativistic quantum mechanics.…