Description of the spectrum of the energy operator of quantum-mechanical systems that is invariant with respect to permutations of identical particles (original) (raw)
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This textbook deals with advanced topics in the field of quantum mechanics, material which is usually encountered in a second university course on quantum mechanics. The book, which comprises a total of 15 chapters, is divided into three parts: I. Many-Body Systems, II. Relativistic Wave Equations, and III. Relativistic Fields. The text is written in such a way as to attach importance to a rigorous presentation while, at the same time, requiring no prior knowledge, except in the field of basic quantum mechanics. The inclusion of all mathematical steps and full presentation of intermediate calculations ensures ease of understanding. A number of problems are included at the end of each chapter. Sections or parts thereof that can be omitted in a first reading are marked with a star, and subsidiary calculations and remarks not essential for comprehension are given in small print. It is not necessary to have read Part I in order to understand Parts II and III. References to other works in the literature are given whenever it is felt they serve a useful purpose. These are by no means complete and are simply intended to encourage further reading. A list of other textbooks is included at the end of each of the three parts.
QUANTUM MECHANICS REVISITED (v.3)
HAL (Le Centre pour la Communication Scientifique Directe), 2015
The paper proposes a new approach in the foundations of Quantum Mechanics. It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models. Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...). It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations. Moreover this approach sheds a new ligth on the issues of determism, and interacting systems. In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
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.
American Journal of Physics, 2007
We discuss an operator solution for the bound states of the non-relativistic hydrogen atom. The method adds the phase of a state and its associated operator to the set of variables of the system. The augmented set of operators is found to form a closed set of commutation relations thus comprising an operator Lie algebra. From these relations, the energy spectrum and bounded radial eigenfunctions are calculated. Our approach is analogous to the one employed to compute the angular momentum spectrum and eigenfunctions but with operators satisfying an su͑1,1͒ Lie algebra instead of su͑2͒. This method, with the same operator algebra and minor modifications, may be used to solve the Dirac relativistic hydrogen atom.
Modern Vibrational Spectroscopy and Micro-Spectroscopy, 2015
Real quantum mechanical systems have the tendency to become mathematically quite complicated and may discourage a novice in the field from pursuing the detailed steps to understand how the mathematical principles apply to physical systems. Thus, a simple scenario is presented here to illustrate the principles of Quantum Mechanics introduced in Section 1.4. The model to be presented is the so-called particle-in-a-box (henceforth referred to as the "PiB") that is an artificial system, yet with wide-ranging analogies to real systems. This model is very instructive, because it shows in detail how the quantum mechanical formalism works in a situation that is sufficiently simple to carry out the calculations step by step. Furthermore, the symmetry (parity) of the PiB wavefunctions is very similar to that of vibrational wavefunctions discussed in Section 1.4. Finally, the concept of transition from one stationary state to another can be demonstrated using the principles of the transition moment introduced in Section 1.5. A.1 Definition of the Model System The PiB model assumes a particle, such as an electron, to be placed into a potential energy well, or confinement shown in Figure A.1. This confinement (the "box") has zero potential energy for 0 ≤ x ≤ L, where L is the length of the box. Outside the box, that is, for x < 0 and for x > L, the potential energy is assumed to be infinite. Thus, once the electron is placed inside the box, it has no chance to escape, and one knows for certain that the electron is in the box. Next, the kinetic and potential energy expression will be defined, which subsequently allows writing the Hamiltonian, or the total energy operator of the system. For any quantum mechanical system, the total energy is written as the sum of the kinetic and potential energies, T and V, respectively:
Coulomb problem in NC quantum mechanics: Exact solution andnon-perturbative aspects
2013
The aim of this paper is to find out how would possible space non-commutativity (NC) alter the QM solution of the Coulomb problem. The NC parameter λ is to be regarded as a measure of the non-commutativity-setting λ = 0 means a return to the standard quantum mechanics. As the very first step a rotationaly invariant NC space R 3 λ , an analog of the Coulomb problem configuration space R 3 0 = R 3 \ {0}, is introduced. R 3 λ is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space F. The properly weighted Hilbert-Schmidt operators in F form H λ , an NC analog of the Hilbert space of the wave functions. We will refer to them as "wave functions" also in the NC case. The definition of an NC analog of the hamiltonian as a hermitian operator in H λ is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for E < 0 and low-energy scattering for E > 0 (both containig NC corrections analytic in λ) and also formulas for high-energy scattering and unexpected bound states at ultra-high energy (both containing NC corrections singular in λ). All the NC contributions to the known QM solutions either vanish or disappear in the limit λ → 0.