A short course on quantum mechanics and methods of quantization

2015 ◽  
Vol 12 (08) ◽  
pp. 1560008 ◽  
Author(s):  
Elisa Ercolessi
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
International Workshop ◽  
Classical Mechanics ◽  
Theoretical Physics ◽  
Feynman Path Integral ◽  
Introductory Courses ◽  
Feynman Path ◽  
Geometric Structures ◽  
Short Course

These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.

scholarly journals p-Adic Path Integrals for Quadratic Actions

1997 ◽  
Vol 12 (20) ◽  
pp. 1455-1463 ◽  
Author(s):  
G. S. Djordjević ◽  
B. Dragovich
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Path Integrals ◽  
Probability Amplitude ◽  
Exact Form ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
One Dimensional ◽  
Ordinary Quantum

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude [Formula: see text] for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.

STATIONARY PHASE, QUANTUM MECHANICS AND SEMI-CLASSICAL LIMIT

1996 ◽  
Vol 08 (08) ◽  
pp. 1161-1185 ◽  
Author(s):  
JORGE REZENDE
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Stationary Phase ◽  
Finite Number ◽  
Path Integral ◽  
Critical Points ◽  
Classical Limit ◽  
Oscillatory Integral ◽  
Feynman Path Integral ◽  
Feynman Path

A method of stationary phase for the normalized-oscillatory integral on Hilbert space is developed in the case where the phase function has a finite number of critical points which are non-degenerate. Applications to the Feynman path integral and the semi-classical limit of quantum mechanics are given.

Exploring Classical Mechanics

2020 ◽  
Author(s):  
G. L. Kotkin ◽  
V. G. Serbo
Keyword(s):  
Quantum Mechanics ◽  
Statistical Mechanics ◽  
Classical Mechanics ◽  
Theoretical Physics ◽  
English Edition ◽  
Mechanical Part ◽  
Russian Edition ◽  
The Given

This book was written by the working physicists for students and teachers of physics faculties of universities. Its contents correspond roughly to the corresponding course in the textbooks Mechanics by L. D. Landau and E. M. Lifshitz (1976) and Classical Mechanics by H. Goldstein, Ch. Poole, and J. Safko (2000). As a rule, the given solution of a problem is not finished with obtaining the required formulae. It is necessary to analyse the results, and this is of great interest and by no means a mechanical part of the solution. The authors consider classical mechanics as the first chapter of theoretical physics; the methods and ideas developed in this chapter are literally important for all other sections of theoretical physics. Thus, the authors have indicated wherever this does not require additional amplification, the analogy or points of contact with the problems in quantum mechanics, electrodynamics, or statistical mechanics. The first English edition of this book was published by Pergamon Press in 1971 with the invaluable help by the translation editor D. ter Haar. This second English publication is based on the fourth Russian edition of 2010 as well as the problems added in the publications in Spanish and French. As a result, this book contains 357 problems instead of the 289 problems that appeared in the first English edition.

The Schrödinger Equation, Path Integration and Applications

2021 ◽  
Vol 15 (01) ◽  
pp. 61-75
Author(s):  
Everaldo M. Bonotto ◽  
Felipe Federson ◽  
Márcia Federson
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Path Integration ◽  
Schrodinger Equation ◽  
Feynman Integral ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Henstock Integral ◽  
The Mean

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

scholarly journals EMERGENT QUANTUM MECHANICS AS A CLASSICAL, IRREVERSIBLE THERMODYNAMICS

2013 ◽  
Vol 10 (04) ◽  
pp. 1350007 ◽  
Author(s):  
D. ACOSTA ◽  
P. FERNÁNDEZ DE CÓRDOBA ◽  
J. M. ISIDRO ◽  
J. L. G. SANTANDER
Keyword(s):  
Quantum Mechanics ◽  
Degrees Of Freedom ◽  
Probability Distributions ◽  
Semiclassical Approximation ◽  
Irreversible Thermodynamics ◽  
Feynman Path Integral ◽  
Fundamental Theory ◽  
Feynman Path ◽  
Additional Support ◽  
Effective Description

We present an explicit correspondence between quantum mechanics and the classical theory of irreversible thermodynamics as developed by Onsager, Prigogine et al. Our correspondence maps irreversible Gaussian Markov processes into the semiclassical approximation of quantum mechanics. Quantum-mechanical propagators are mapped into thermodynamical probability distributions. The Feynman path integral also arises naturally in this setup. The fact that quantum mechanics can be translated into thermodynamical language provides additional support for the conjecture that quantum mechanics is not a fundamental theory but rather an emergent phenomenon, i.e. an effective description of some underlying degrees of freedom.

NELSON'S STOCHASTIC MECHANICS BY MEANS OF FEYNMAN PATH INTEGRALS

2000 ◽  
Vol 14 (03) ◽  
pp. 73-78 ◽  
Author(s):  
LUIZ C. L. BOTELHO
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Path Integrals ◽  
Order Equation ◽  
Feynman Path Integral ◽  
Stochastic Mechanics ◽  
Path Integral Formulation ◽  
Feynman Path ◽  
First Order ◽  
Feynman Path Integrals

We show that Nelson's stochastic mechanics suitably formulated as a Hamilton–Jacobi first-order equation leads straightforwardly to the Feynman path integral formulation of quantum mechanics.

scholarly journals Ultra-light and strong: The massless harmonic oscillator and its singular path integral

2016 ◽  
Vol 14 (01) ◽  
pp. 1750010
Author(s):  
Giovanni Modanese
Keyword(s):  
Harmonic Oscillator ◽  
Path Integral ◽  
High Frequency ◽  
Classical Mechanics ◽  
Planck Equation ◽  
Initial Position ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Limit Formula ◽  
Hilbert Action

In classical mechanics, a light particle bound by a strong elastic force just oscillates at high frequency in the region allowed by its initial position and velocity. In quantum mechanics, instead, the ground state of the particle becomes completely de-localized in the limit [Formula: see text]. The harmonic oscillator thus ceases to be a useful microscopic physical model in the limit [Formula: see text], but its Feynman path integral has interesting singularities which make it a prototype of other systems exhibiting a “quantum runaway” from the classical configurations near the minimum of the action. The probability density of the coherent runaway modes can be obtained as the solution of a Fokker–Planck equation associated to the condition [Formula: see text]. This technique can be applied also to other systems, notably to a dimensional reduction of the Einstein–Hilbert action.

scholarly journals Prediction of Unified New Physics beyond Quantum Mechanics from the Feynman Path Integral: Elementary Cycles String Theory

2021 ◽  
Author(s):  
Donatello Dolce
Keyword(s):  
Path Integral ◽  
World Line ◽  
Internal Clock ◽  
New Physics ◽  
Theoretical Physics ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Quantum Phenomena ◽  
Time Accuracy

Abstract We prove that the Feynman Path Integral is equivalent to a novel stringy description of elementary particles characterized by a single compact (cyclic) world-line parameter playing the role of the particle internal clock. This clearly reveals an exact unified formulation of quantum and relativistic physics, potentially deterministic, fully falsifiable having no fine-tunable parameters, also proven in previous pap,rs to be completely consistent with all known physics, from theoretical physics to condensed matter. New physics will be discovered by observing quantum phenomena with experimental time accuracy of the order of 10-2 sec.

ALTERNATIVE HAMILTONIAN DESCRIPTIONS FOR QUANTUM SYSTEMS AND NON-HERMITIAN OPERATORS WITH REAL SPECTRUM

2002 ◽  
Vol 17 (24) ◽  
pp. 1589-1599 ◽  
Author(s):  
FRANCO VENTRIGLIA
Keyword(s):  
Inverse Problem ◽  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Systems ◽  
Theoretical Physics ◽  
Inner Product ◽  
Real Spectrum ◽  
Standard Methods ◽  
Finite Dimensional

Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

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