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Mathematical engineering

Tensors and Bra-Ket Notation in Quantum Mechanics

Hung Nguyen-Schäfer, Jan-Philip Schmidt

Published Aug 16, 2016• 2 citations
Publisher

Abstract

Classical mechanics have been used to study Newtonian mechanics including conventional mechanics, electrodynamics, cosmology, and relativity physics of Einstein. Compared to quantum mechanics, classical mechanics is deterministic, in which the processes are well determined and foreseeable. On the contrary, processes in quantum mechanics are undeterministic, uncertain, nonlocal, and intrinsically random. Therefore, quantum mechanics deals with probability and uncertainty principles to study the atomic and subatomic world. Due to their extremely small sizes of the quantum particles, the capabilities of apparatuses are quite limited to measure the undeterministic and uncertain behaviors of the particles. Therefore, interactions between these very small particles in the quantum world, such as electrons and photons are simulated using mathematical abstractions that will be discussed in the following sections. Mathematics of quantum physics describes nonlocal correlations between the particles with the intrinsic randomness (e.g. Schrödinger’s cat) in the combined system. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-662-48497-5_6

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Newtonian mechanics is the foundation of Classical Physics. Newton’s mechanics, Thermodynamics, Wave theory of Optics and Maxwell’s electromagnetic theory belong to regime of Classical Physics and can be used to explain a wide range of phenomena in the Universe at macroscopic scale. These theories fail spectacularly when applied to phenomenain the atomic and nuclear regime, for example, proton-atom scattering or the flow of electrons in a semiconductor. Quantum mechanics is the most successful scientific theory that has ever been created and it has completely changed our view of the world. The failure of Classical Physics was highlighted by black body radiation and photoelectric effect. Max Planck and Albert Einstein provided explanations of both phenomena based on quantum hypothesis and are thus considered founders of Quantum Physics. The origin of quantum mechanics goes back to the mid-1920s.It was formulated first as matrix mechanics by Werner Heisenberg, Max Born and Pascual Jordan; then as wave mechanics by Louis de Broglie and Erwin Schrödinger; and later on asquantum statistics of subatomic particles by Fermi-Dirac and Bose-Einstein. Combining relativistic mechanics with quantum mechanics, Dirac formulated his relativistic quantum mechanics during 1930s. Uncertainty Principle is thecornerstone of Quantum Physics. The role of randomness in microscopic physical processes shatters the myth that the universe is deterministic. Quantum world is unpredictable in the classical sense and demolishes the idea of an objective universe. The Copenhagen interpretation remains the quantum mechanical formalism that is currently most widely accepted amongst physicists.Quantum theories support cosmic spirit pervading the cosmos and inter – relationship of individuals in the world society. Quantum philosophy is holistic and is going to revolutionize our world-view.
In recent years decoherence has been widely discussed and accepted as a mechanism responsible for the emergence of classicality in quantum measurements and the absence in the real world of Schrödinger-cat-like states [ 2 , 6 , 7 , 11 , 12 ]. Let me start with recalling the Schrödinger cat paradox. It was a hypothetical experiment proposed by Schrödinger in 1935 which can be briefly described as follows. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Quantum mechanics (or simply quantum physics) is the branch of physics that applies to particles and systems of particles, such as atoms and molecules. In that realm the laws of classical mechanics and classical electromagnetism break down, and only quantum mechanics provides the concepts and computational methods that explain observed phenomena. Quantum mechanics uses specially developed tools, such as wave functions, that contain all of the relevant information about particles and systems. is approach is needed to understand the true nature of electromagnetic radiation and how that radiation interacts with matter. e mathematics used in quantum mechanics is highly specialized and often complex. In this chapter we’ll address some of the key questions that require quantum mechanics without presenting the full mathematical formalism. e important questions we’ll address all deal with observable physical phenomena. We’ll discuss how quantum mechanics is used to answer these questions but will omit the advanced mathematical details.