A squeezed state holomorphic phase space representation of equations of motion

A mapping scheme is presented which takes quantum operators associated to bosonic degrees of freedom into complex phase space integral kernel representatives. The procedure consists of using the Schrödinger squeezed state as the starting point for the construction of the integral mapping kernel which, due to its inherent structure, is suited for the description of second quantized operators. Products and commutators of operators have their representatives explicitly written which reveal new details when compared to the usual q-p phase space description. The classical limit of the equations of motion for the canonical pair q-p is discussed in connection with the effect of squeezing the quantum phase space cellular structure. © 1993.

General method for reducing the two-body Dirac equation

A semirelativistic two-body Dirac equation with an enlarged set of phenomenological potentials, including Breit-type terms, is investigated for the general case of unequal masses. Solutions corresponding to definite total angular momentum and parity are shown to fall into two classes, each one being obtained by solving a system of four coupled first-order radial differential equations. The reduction of each of these systems to a pair of coupled Schrödinger-type equations is also discussed. © 1992 American Institute of Physics.

Evaluation of effective nucleon-nucleon potential and description of α-cluster type nuclei in restricted dynamics approach

The application of the Restricted Dynamics Approach in nuclear theory, based on the approximate solution of many-particle Schrödinger equation, which accounts for all conservation laws in many-nucleon system, is discussed. The Strictly Restricted Dynamics Model is used for the evaluation of binding energies, level schemes, E2 and Ml transition probabilities as well as the electric quadrupole and magnetic dipole momenta of light a-cluster type nuclei in the region 4 ≤ A ≤ 40. The parameters of effective nucleonnucleon interaction potential are evaluated from the ground state binding energies of doubly magic nuclei 4He, 16O and 40Ca.

Eletrodinâmica de ordem superior em (2+1)D

Pós-graduação em Física - IFT

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Estudo de sistemas não-lineares em mecânica quântica

The main goal of this work is to investigate the effects of a nonlinear cubic term inserted in the Schrödinger equation for one-dimensional potentials studied in Quantum Mechanics textbooks. Being the main tool the numerical analysis in a large number of works, the analysis of this effect by this term in the potential itself, in order to work with an analytical solution, can be considered something new. For the harmonic oscillator potential, the analysis was made from a numerical method, comparing the result with the known results in the literature. In the case of the infinite well potential and the step potential, hoping to work with an analytical solution, by construction we started with the known wavefunction for the linear case noting the effects in the other physical quantities. The coupling of the physical quantities involved in this work has yielded, besides many complications in the calculations, a series of conditions on the existence and validity of the solutions in regard to the system possible configurations

Few-cycle attosecond pulse chirp effects on asymmetries in ionized electron momentum distributions

The momentum distributions of electrons ionized from H atoms by chirped few-cycle attosecond pulses are investigated by numerically solving the time-dependent Schrödinger equation. The central carrier frequency of the pulse is chosen to be 25 eV, which is well above the ionization threshold. The asymmetry (or difference) in the yield of electrons ionized along and opposite to the direction of linear laser polarization is found to be very sensitive to the pulse chirp (for pulses with fixed carrier-envelope phase), both for a fixed electron energy and for the energy-integrated yield. In particular, the larger the pulse chirp, the larger the number of times the asymmetry changes sign as a function of ionized electron energy. For a fixed chirp, the ionized electron asymmetry is found to be sensitive also to the carrier-envelope phase of the few-cycle pulse.

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Estudo de sistemas não-lineares em mecânica quântica

The main goal of this work is to investigate the effects of a nonlinear cubic term inserted in the Schrödinger equation for one-dimensional potentials studied in Quantum Mechanics textbooks. Being the main tool the numerical analysis in a large number of works, the analysis of this effect by this term in the potential itself, in order to work with an analytical solution, can be considered something new. For the harmonic oscillator potential, the analysis was made from a numerical method, comparing the result with the known results in the literature. In the case of the infinite well potential and the step potential, hoping to work with an analytical solution, by construction we started with the known wavefunction for the linear case noting the effects in the other physical quantities. The coupling of the physical quantities involved in this work has yielded, besides many complications in the calculations, a series of conditions on the existence and validity of the solutions in regard to the system possible configurations

Resonant States in Negative Ions

Resonant states are multiply excited states in atoms and ions that have enough energy to decay by emitting an electron. The ability to emit an electron and the strong electron correlation (which is extra strong in negative ions) makes these states both interesting and challenging from a theoretical point of view. The main contribution in this thesis is a method, which combines the use of B splines and complex rotation, to solve the three-electron Schrödinger equation treating all three electrons equally. It is used to calculate doubly excited and triply excited states of 4S symmetry with even parity in He-. For the doubly excited states there are experimental and theoretical data to compare with. For the triply excited states there is only theoretical data available and only for one of the resonances. The agreement is in general good. For the triply excited state there is a significant and interesting difference in the width between our calculation and another method. A cause for this deviation is suggested. The method is also used to find a resonant state of 4S symmetry with odd parity in H2-. This state, in this extremely negative system, has been predicted by two earlier calculations but is highly controversial. Several other studies presented here focus on two-electron systems. In one, the effect of the splitting of the degenerate H(n=2) thresholds in H-, on the resonant states converging to this threshold, is studied. If a completely degenerate threshold is assumed an infinite series of states is expected to converge to the threshold. Here states of 1P symmetry and odd parity are examined, and it is found that the relativistic and radiative splitting of the threshold causes the series to end after only three resonant states. Since the independent particle model completely fails for doubly excited states, several schemes of alternative quantum numbers have been suggested. We investigate the so called DESB (Doubly Excited Symmetry Basis) quantum numbers in several calculations. For the doubly excited states of He- mentioned above we investigate one resonance and find that it cannot be assigned DESB quantum numbers unambiguously. We also investigate these quantum numbers for states of 1S even parity in He. We find two types of mixing of DESB states in the doubly excited states calculated. We also show that the amount of mixing of DESB quantum numbers can be inferred from the value of the cosine of the inter-electronic angle. In a study on Li- the calculated cosine values are used to identify doubly excited states measured in a photodetachment experiment. In particular a resonant state that violates a propensity rule is found.

Quantum hydrodynamic models for semiconductors with and without collisions

In dieser Arbeit werden Quantum-Hydrodynamische (QHD) Modelle betrachtet, die ihren Einsatz besonders in der Modellierung von Halbleiterbauteilen finden. Das QHD Modell besteht aus den Erhaltungsgleichungen für die Teilchendichte, das Momentum und die Energiedichte, inklusive der Quanten-Korrekturen durch das Bohmsche Potential. Zu Beginn wird eine Übersicht über die bekannten Ergebnisse der QHD Modelle unter Vernachlässigung von Kollisionseffekten gegeben, die aus ein­em Schrödinger-System für den gemischten-Zustand oder aus der Wigner-Glei­chung hergeleitet werden können. Nach der Reformulierung der eindimensionalen QHD Gleichungen mit linearem Potential als stationäre Schrö­din­ger-Gleichung werden die semianalytischen Fassungen der QHD Gleichungen für die Gleichspannungs-Kurve betrachtet. Weiterhin werden die viskosen Stabilisierungen des QHD Modells be­rück­sich­tigt, sowie die von Gardner vorgeschlagene numerische Viskosität für das {sf upwind} Finite-Differenzen Schema berechnet. Im Weiteren wird das viskose QHD Modell aus der Wigner-Glei­chung mit Fokker-Planck Kollisions-Ope­ra­tor hergeleitet. Dieses Modell enthält die physikalische Viskosität, die durch den Kollision-Operator eingeführt wird. Die Existenz der Lösungen (mit strikt positiver Teilchendichte) für das isotherme, stationäre, eindimensionale, viskose Modell für allgemeine Daten und nichthomogene Randbedingungen wird gezeigt. Die dafür notwendigen Abschätzungen hängen von der Viskosität ab und erlauben daher den Grenzübergang zum nicht-viskosen Fall nicht. Numerische Simulationen der Resonanz-Tunneldiode modelliert mit dem nichtisothermen, stationären, eindimensionalen, viskosen QHD Modell zeigen den Einfluss der Viskosität auf die Lösung. Unter Verwendung des von Degond und Ringhofer entwickelten Quanten-Entropie-Minimierungs-Verfahren werden die allgemeinen QHD-Gleichungen aus der Wigner-Boltzmann-Gleichung mit dem BGK-Kollisions-Operator hergeleitet. Die Herleitung basiert auf der vorsichtige Entwicklung des Quanten-Max­well­ians in Potenzen der skalierten Plankschen Konstante. Das so erhaltene Modell enthält auch vertex-Terme und dispersive Terme für die Ge­schwin­dig­keit. Dadurch bleibt die Gleichspannungs-Kurve für die Re­so­nanz-Tunnel­diode unter Verwendung des allgemeinen QHD Modells in einer Dimension numerisch erhalten. Die Ergebnisse zeigen, dass der dispersive Ge­schwin­dig­keits-Term die Lösung des Systems stabilisiert.

Constructing correlated spin states with neutral atoms in optical lattices

This thesis reports on the experimental investigation of controlled spin dependent interactions in a sample of ultracold Rubidium atoms trapped in a periodic optical potential. In such a situation, the most basic interaction between only two atoms at one common potential well, forming a micro laboratory for this atom pair, can be investigated. Spin dependent interactions between the atoms can lead to an intriguing time evolution of the system. In this work, we present two examples of such spin interaction induced dynamics. First, we have been able to observe and control a coherent spin changing interaction. Second, we have achieved to examine and manipulate an interaction induced time evolution of the relative phase of a spin 1/2-system, both in the case of particle pairs and in the more general case of N interacting particles. The first part of this thesis elucidates the spin-changing interaction mechanism underlying many fascinating effects resulting from interacting spins at ultracold temperatures. This process changes the spin states of two colliding particles, while preserving total magnetization. If initial and final states have almost equal energy, this process is resonant and leads to large amplitude oscillations between different spin states. The measured coupling parameters of such a process allow to precisely infer atomic scattering length differences, that e.g. determine the nature of the magnetic ground state of the hyperfine states in Rubidium. Moreover, a method to tune the spin oscillations at will based on the AC-Zeeman effect has been implemented. This allowed us to use resonant spin changing collisions as a quantitative and non-destructive particle pair probe in the optical lattice. This led to a series of experiments shedding light on the Bosonic superfluid to Mott insulator transition. In a second series of experiments we have been able to coherently manipulate the interaction induced time evolution of the relative phase in an ensemble of spin 1/2-systems. For two particles, interactions can lead to an entanglement oscillation of the particle pair. For the general case of N interacting particles, the ideal time evolution leads to the creation of spin squeezed states and even Schrödinger cat states. In the experiment we have been able to control the underlying interactions by a Feshbach resonance. For particle pairs we could directly observe the entanglement oscillations. For the many particle case we have been able to observe and reverse the interaction induced dispersion of the relative phase. The presented results demonstrate how correlated spin states can be engineered through control of atomic interactions. Moreover, the results point towards the possibility to simulate quantum magnetism phenomena with ultracold atoms in optical traps, and to realize and analyze many novel quantum spin states which have not been experimentally realized so far.

Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

Resonances in the two centers Coulomb system

In this work we investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges in two and three dimensions, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrödinger operator. After giving a description of the bifurcation of the classical system for positive energies, we construct the resolvent kernel of the operators and we prove that they can be extended analytically to the second Riemann sheet. The resonances are then defined and studied with numerical methods and perturbation theory.

An infinite level atom coupled to a heat bath

Wir untersuchen die Mathematik endlicher, an ein Wärmebad gekoppelter Teilchensysteme. Das Standard-Modell der Quantenelektrodynamik für Temperatur Null liefert einen Hamilton-Operator H, der die Energie von Teilchen beschreibt, welche mit Photonen wechselwirken. Im Heisenbergbild ist die Zeitevolution des physikalischen Systems durch die Wirkung einer Ein-Parameter-Gruppe auf eine Menge von Observablen A gegeben: Diese steht im Zusammenhang mit der Lösung der Schrödinger-Gleichung für H. Um Zustände von A, welche das physikalische System in der Nähe des thermischen Gleichgewichts zur Temperatur T darstellen, zu beschreiben, folgen wir dem Ansatz von Jaksic und Pillet, eine Darstellung von A zu konstruieren. Die Vektoren in dieser Darstellung definieren die Zustände, die Zeitentwicklung wird mit Hilfe des Standard Liouville-Operators L beschrieben. In dieser Doktorarbeit werden folgende Resultate bewiesen bzw. hergeleitet: - die Konstuktion einer Darstellung - die Selbstadjungiertheit des Standard Liouville-Operators - die Existenz eines Gleichgewichtszustandes in dieser Darstellung - der Limes des physikalischen Systems für große Zeiten.