Principles of Quantum Mechanics SECOND EDITION
WILLIAM ARI SANTAMARIA CASTRO
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Abstract
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This comprehensive work on quantum mechanics delves into the foundational principles and practical applications of the subject. It emphasizes the importance of understanding postulates and operator assignments, while guiding students through complex concepts such as scattering, path integrals, symmetries, and the Dirac equation. The pedagogical approach balances theoretical rigor with accessibility, aiming to prepare students for advanced studies in physics.
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CHAPTER 4 Introduction to Quantum Mechanics
183 philosophical implications of quantum mechanics and develop a new way of thinking about nature on the nanometer-length scale. This was undoubtedly one of the most signiicant shifts in the history of science. The key new concepts developed in quantum mechanics include the quantiza-tion of energy, a probabilistic description of particle motion, wave–particle duality, and indeterminacy. These ideas appear foreign to us because they are inconsistent with our experience of the macroscopic world. Nonetheless, we have accepted their validity because they provide the most comprehensive account of the behavior of matter and radiation and because the agreement between theory and the results of all experiments conducted to date has been impressively accurate. Energy quantization arises for all systems whose motions are connned by a potential well. The one-dimensional particle-in-a-box model shows why quantiza-tion only becomes apparent on the atomic scale. Because the energy level spacing is inversely proportional to the mass and to the square of the length of the box, quantum effects become too small to be observed for systems that contain more than a few hundred atoms. Wave–particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, we can form analogies with classical electromagnetic wave theory to describe the motion of particles. For example, the probability of locating the particle at a particular location is the square of the amplitude of its wave function. Zero-point energy is a consequence of the Heisenberg indeterminacy relation; all particles bound in potential wells have nite energy even at the absolute zero of temperature. Particle-in-a-box models illustrate a number of important features of quantum mechanics. The energy-level structure depends on the nature of the potential, E n n 2 , for the particle in a one-dimensional box, so the separation between energy levels increases as n increases. The probability density distribution is different from that for the analogous classical system. The most probable location for the particle-in-a-box model in its ground state is the center of the box, rather than uniformly over the box as predicted by classical mechanics. Normalization ensures that the probability of nding the particle at some position in the box, summed over all possible positions, adds up to 1. Finally, for large values of n, the probability distribution looks much more classical, in accordance with the correspondence principle. Different kinds of energy level patterns arise from different potential energy functions, for example the hydrogen atom (See Section 5.1) and the harmonic oscil-lator (See Section 20.3). These concepts and principles are completely general; they can be applied to explain the behavior of any system of interest. In the next two chapters, we use quantum mechanics to explain atomic and molecular structure, respectively. It is important to have a rm grasp of these principles because they are the basis for our comprehensive discussion of chemical bonding in Chapter 6.
Introduction to Quantum Mechanics
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A Concise Treatise on Quantum Mechanics in Phase Space
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Wigner's quasi-probability distribution function in phase-space is a special (Weyl-Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics, nuclear physics, quantum computing, decoherence, and chaos. It is also of importance in signal processing, and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory: The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways. This treatise provides an introductory overview and includes an extensive bibliography. Still, the bibliography makes no pretense to exhaustiveness. The overview collects often-used practical formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. As a concise treatise, it provides supplementary material which may be used for an advanced undergraduate or a beginning graduate course in quantum mechanics. It represents an expansion of a previous overview with selected papers collected by the authors, and includes a historical narrative account due the subject. This Historical Survey is presented first, in Section 1, but it might be skipped by students more anxious to get to the mathematical details beginning with the Introduction in Section 2. Alternatively, Section 1 may be read alone by anyone interested only in the history of the subject. Peter Littlewood and Harry Weerts are thanked for allotting time to make the treatise better.
Quantum Theory 101
By the end of the nineteenth century theoretical physicists thought that soon they could pack up their bags and go home. They had developed a powerful mathematical theory, classical mechanics, which seemed to described just about all that they observed, with the exception of a few sticking points. In particular the classical world was ruled by Newtonian physics where matter (atoms) interacted with a radiation field (light) as described by Maxwell's equations. However as it turned out these sticking points were not smoothed over at all but rather were glimpses of the microscopic (and relativistic) world which was soon to be experientially discovered. As has been stated countless times, by the end of the first decade of the twentieth century quantum mechanics and relatively had appeared and would soon cause classical mechanics, with its absolute notions of space, time and determinacy to be viewed as an approximation. Historically the first notion of quantized energy came in 1900 with Planck's hypothesis that the energy contained in radiation could only be exchanged in discrete lumps, known as " quanta " , with matter. This in turn implies that the energy, E, of radiation is proportional to its frequency, ν, E = hν (1.1) with h a constant, Planck's constant. This allowed him to derive his famous formula for black body spectra which was in excellent agreement with experiment. From our perspective this is essentially a thermodynamic issue and the derivation is therefore out of the main theme of this course. The derivation of such a formula from first principles was one of the sticking points mentioned above. While Planck's derivation is undoubtedly the first appearance of quantum ideas it was not at all clear at the time that this was a fundamental change, or that there was some underlying classical process which caused the discreteness, or that it was even correct. However there were further experiments which did deeply challenge the continuity of the world and whose resolution relies heavy on Planck's notion of quanta.
Student Difficulties with Quantum Mechanics Formalism
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We discuss student difficulties in distinguishing between the physical space and Hilbert space and difficulties related to the Time-independent Schroedinger equation and measurements in quantum mechanics. These difficulties were identified by administering written surveys and by conducting individual interviews with students.
A New Approach to Understanding Quantum Mechanics: Illustrated Using a Pedagogical Model over ℤ2
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