Quantum Harmonic Oscillator

In this paper, I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be described in terms of classical physics without invoking violations of the energy conservation law at any time instance. A formula is presented that generates a wide class of potential barrier shapes with the desirable reflection (transmission) coefficient and transmission phase shift along with the corresponding exact solutions of the time-independent Schr\"odinger's equation. These results, with support from numerical simulations, strongly suggest that two uncoupled classical harmonic oscillators, which initially have a 90\degree relative phase shift and then are simultaneously disturbed by the same parametric perturbation of a finite duration, manifest behavior which can be mapped to that of a single quantum particle, with classical 'ran...

The Caldeira-Leggett Hamiltonian describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas (Landau, 1946 [1]). An invertible linear transformation (Morrison and Pfirsch, 1992 [18]; Morrison, 2000 [19]) is presented that converts solutions of the Caldeira-Leggett model into solutions of the linearized Vlasov-Poisson system.

An exact quantization rule for the Schr\"{o}dinger equation is presented. In the exact quantization rule, in addition to $N\pi$, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second P\"{o}schl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom.

The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = phi and Q = -1/phi, and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n --> ∞ it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated.

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ``shape-invariant'' potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

This paper presents a low dimensional topological approach to the formation of different Quantum Particles by Geometric Quantization of the energy excited localized harmonic oscillation of the field. Establishing a single function for both Bosons and Fermions. Further derived, is the formulation for calculating physical properties, such as mass and size which coincides with the known experimental values. As a natural consequence, Strong and Weak nuclear forces manifest as higher order Electromagnetic force, moving closer towards unification and leading to possibilities for further higher order forces. Physical interpretation of De-Broglie Wavelength, Compton Wavelength, Wave-Particle Duality, Quantum Spin and particle’s internal structure are established along with explanation for phenomena like Quantum Locking and Neutrino Oscillation.

The Schwinger representation is an elegant formulation in QM where the theory of angular momentum and harmonic oscillator are nicely blended. In the direct product space of kets, there exists a subspace confined to the angular momentum operators created suitably from the creation and annihilation operators of SHO.

All'interno del lavoro viene mostrato come la rotazione di Wick e il formalismo del tempo immaginario permettano in maniera semplice di identificare lo stato fondamentale di un sistema quantistico. Vengono poi sviluppate due tecniche di risoluzione numerica basate su questo formalismo: lo splitting degli operatori e la diffusione montecarlo.
Si descrivere poi un metodo di integrazione numerica dell'equazione di Schrödinger  indipendente dal tempo, l'algoritmo di Numerov, associato alla procedura di shooting.
I tre metodi vengono infine utilizzati per studiare la buca di potenziale infinita, l'oscillatore armonico e l'oscillatore quartico.

This is a graduate course on elementary quantum mechanics written for the benefit of undergraduate and graduate students. It is the English version of physics/0003106, which I did at the suggestion of several students from different countries. The topics included refer to the postulates of quantum mechanics, one-dimensional barriers and wells, angular momentum and spin, WKB method, harmonic oscillator, hydrogen atom, quantum scattering, and partial waves

This is the first Internet course on elementary quantum mechanics written in Spanish ("castellano") for the benefit of Spanish speaking students. I thank my eight Mexican students at the Institute of Physics, University of Guanajuato, Leon, (IFUG), for the collaboration that made this possible. The topics included refer to the postulates of quantum mechanics, one-dimensional barriers and wells, angular momentum,

Bu pdf fizik (veya astronomi, matematik) lisans öğrencilerine kuantum mekaniğinin kullandığı matematiğin göründüğü kadar karmaşık olmadığını göstermek amacıyla Cohen ve Shankar'ı kaynak alarak oluşturduğum yaklaşık 80 sayfalık bir nottur. Parçacıkları artık doğrudan üç konum ve üç momentum ile tanımlamak (ara ara klasik mekaniğe de değiniyorum) yerine bir dalga fonksiyonu ile tanımlayarak başlayıp küresel harmonikler ile bitiriyorum. Spin operatörü için kuantum mekaniği II'de devam edeceğim.

It has been recently investigated that the jerk dynamical systems are the simplest ever systems, which possess variety of dynamical behaviors including chaotic motion. Interestingly, the jerk dynamical systems also describe various phenomena in physics and engineering such as electrical circuits, mechanical oscillators, laser physics, solar wind driven magnetosphere ionosphere (WINDMI) model, damped harmonic oscillator driven by nonlinear memory term, biological systems etc. In many practical situations chaos is undesirable phenomenon, which may lead to irregular operations in physical systems. Thus from a practical point of view, one would like to convert chaotic solutions into periodic limit cycle or fixed point solutions. On the other hand, there has been growing interest to use chaos profitably by synchronizing chaotic systems due to its potential applications in secure communication. In this paper, we have made a thorough investigation of synchronization of identical chaotic jerk dynamical systems by implementing three well-known techniques: (i) Pecora-Carroll (PC) technique, (ii) Feedback (FB) technique and (iii) Active Passive decomposition (APD). We have given a detailed review of these techniques followed by the results of our investigations of identical synchronization of chaos in jerk dynamical systems. The stability of identical synchronization in all the aforesaid methods has also been discussed through the transversal stability analysis. Our extensive numerical calculation results reveal that in PC and FB techniques the x-drive configuration is able to produce the stable identical synchronization in all the chaotic jerk dynamical systems considered by us (except for a few cases), however y-drive and z-drive configurations do not lead to the stable identical synchronization. For the APD approach, we have suggested a generalized active passive decomposition, which leads to the stable identical synchronization without being bothered about the specific form of the jerk dynamical system. Several other active passive decompositions have also been listed with their corresponding conditional Lyapunov exponents to achieve the stable identical synchronization in various chaotic jerk dynamical systems.