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.

Il ruolo della metafora nella comunicazione della fisica contemporanea

Il presente lavoro si rivolge all’analisi del ruolo delle forme metaforiche nella divulgazione della fisica contemporanea. Il focus è sugli aspetti cognitivi: come possiamo spiegare concetti fisici formalmente complessi ad un audience di non-esperti senza ‘snaturarne’ i significati disciplinari (comunicazione di ‘buona fisica’)? L’attenzione è sulla natura stessa della spiegazione e il problema riguarda la valutazione dell’efficacia della spiegazione scientifica a non-professionisti. Per affrontare tale questione, ci siamo orientati alla ricerca di strumenti formali che potessero supportarci nell’analisi linguistica dei testi. La nostra attenzione si è rivolta al possibile ruolo svolto dalle forme metaforiche nella costruzione di significati disciplinarmente validi. Si fa in particolare riferimento al ruolo svolto dalla metafora nella comprensione di nuovi significati a partire da quelli noti, aspetto fondamentale nel caso dei fenomeni di fisica contemporanea che sono lontani dalla sfera percettiva ordinaria. In particolare, è apparsa particolarmente promettente come strumento di analisi la prospettiva della teoria della metafora concettuale. Abbiamo allora affrontato il problema di ricerca analizzando diverse forme metaforiche di particolare rilievo prese da testi di divulgazione di fisica contemporanea. Nella tesi viene in particolare discussa l’analisi di un case-study dal punto di vista della metafora concettuale: una analogia di Schrödinger per la particella elementare. I risultati dell’analisi suggeriscono che la metafora concettuale possa rappresentare uno strumento promettente sia per la valutazione della qualità delle forme analogiche e metaforiche utilizzate nella spiegazione di argomenti di fisica contemporanea che per la creazione di nuove e più efficaci metafore. Inoltre questa prospettiva di analisi sembra fornirci uno strumento per caratterizzare il concetto stesso di ‘buona fisica’. Riteniamo infine che possano emergere altri risultati di ricerca interessanti approfondendo l’approccio interdisciplinare tra la linguistica e la fisica.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Applicazioni della geometria simplettica alla risoluzione delle equazioni iconali

La tesi affronta il problema della risoluzione delle equazioni di tipo iconale, introducendo delle metodologie simplettiche, ovvero tramite l'uso di sottovarietà Lagrangiane. Si guarda nello specifico alla risoluzione dell'equazione agli autovalori di Schrödinger in una e più dimensioni, mostrando la tecnica approssimativa WKB.

Orbitali molecolari di legame e antilegame per lo ione molecolare idrogeno H2+: risoluzione esatta e approssimazione LCAO

In questo lavoro di tesi si intende fornire un'analisi in chiave quantomeccanica di una serie di caratteristiche della molecola di idrogeno ionizzata. Il fatto che l'equazione di Schrödinger per l'elettrone sia nel caso di H2+ risolvibile in maniera esatta rende questo sistema fisico un prezioso banco di prova per qualsiasi metodo di approssimazione. Il lavoro svolto in questa trattazione consisterà proprio nella risoluzione dell'equazione d'onda per l'elettrone nel suo stato fondamentale, dapprima in maniera esatta poi mediante LCAO, e successivamente nell'analisi dei risultati ottenuti, che verranno dapprima discussi e interpretati in chiave fisica, e infine messi a confronto per la verifica della bontà dell'approssimazione. Il metodo approssimato fornirà approssimazioni relative anche al primo stato elettronico eccitato; anche questo verrà ampiamente discusso, e ci si soffermerà in particolare sulla caratterizzazione di orbitali di "legame" e di "antilegame", e sul loro rapporto con la stabilità dello ione molecolare.

Fasi geometriche, fase di Berry ed effetto Aharonov-Bohm

Un sistema sottoposto ad una lenta evoluzione ciclica è descritto da un'Hamiltoniana H(X_1(t),...,X_n(t)) dipendente da un insieme di parametri {X_i} che descrivono una curva chiusa nello spazio di appartenenza. Sotto le opportune ipotesi, il teorema adiabatico ci garantisce che il sistema ritornerà nel suo stato di partenza, e l'equazione di Schrödinger prevede che esso acquisirà una fase decomponibile in due termini, dei quali uno è stato trascurato per lungo tempo. Questo lavoro di tesi va ad indagare principalmente questa fase, detta fase di Berry o, più in generale, fase geometrica, che mostra della caratteristiche uniche e ricche di conseguenze da esplorare: essa risulta indipendente dai dettagli della dinamica del sistema, ed è caratterizzata unicamente dal percorso descritto nello spazio dei parametri, da cui l'attributo geometrico. A partire da essa, e dalle sue generalizzazioni, è stata resa possibile l'interpretazione di nuovi e vecchi effetti, come l'effetto Aharonov-Bohm, che pare mettere sotto una nuova luce i potenziali dell'elettromagnetismo, e affidare loro un ruolo più centrale e fisico all'interno della teoria. Il tutto trova una rigorosa formalizzazione all'interno della teoria dei fibrati e delle connessioni su di essi, che verrà esposta, seppur in superficie, nella parte iniziale.

Stime Integrali su Gruppi di Tipo H e Principio Forte di Continuazione Unica

Lo scopo di questa tesi è dimostrare il Principio Forte di Continuazione Unica per opportune soluzioni di un'equazione di tipo Schrödinger Du=Vu, ove D è il sub-Laplaciano canonico di un gruppo di tipo H e V è un potenziale opportuno. Nel primo capitolo abbiamo esposto risultati già noti in letteratura sui gruppi di tipo H: partendo dalla definizione di tali gruppi, abbiamo fornito un'utile caratterizzazione in termini "elementari" che permette di esplicitare la soluzione fondamentale dei relativi sub-Laplaciani canonici. Nel secondo capitolo abbiamo mostrato una formula di rappresentazione per funzioni lisce sui gruppi di tipo H, abbiamo dimostrato una forma forte del Principio di Indeterminazione di Heisenberg (sempre nel caso di gruppi di tipo H) e abbiamo fornito una formula per la variazione prima dell'integrale di Dirichlet associato a Du=Vu. Nel terzo capitolo, infine, abbiamo analizzato le proprietà di crescita di funzioni di frequenza, utili a dimostrare le stime integrali che implicano in modo piuttosto immediato il Principio Forte di Continuazione Unica, principale oggetto del nostro studio.

Holographic renormalization for z = 2 Lifshitz spacetimes from AdS

Lifshitz spacetimes with the critical exponent z = 2 can be obtained by the dimensional reduction of Schrödinger spacetimes with the critical exponent z = 0. The latter spacetimes are asymptotically AdS solutions of AdS gravity coupled to an axion–dilaton system and can be uplifted to solutions of type IIB supergravity. This basic observation is used to perform holographic renormalization for four-dimensional asymptotically z = 2 locally Lifshitz spacetimes by the Scherk–Schwarz dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically locally AdS spacetimes coupled to an axion–dilaton system. We can thus define and characterize a four-dimensional asymptotically locally z = 2 Lifshitz spacetime in terms of five-dimensional AdS boundary data. In this setup the four-dimensional structure of the Fefferman–Graham expansion and the structure of the counterterm action, including the scale anomaly, will be discussed. We find that for asymptotically locally z = 2 Lifshitz spacetimes obtained in this way, there are two anomalies each with their own associated nonzero central charge. Both anomalies follow from the Scherk–Schwarz dimensional reduction of the five-dimensional conformal anomaly of AdS gravity coupled to an axion–dilaton system. Together, they make up an action that is of the Horava–Lifshitz type with a nonzero potential term for z = 2 conformal gravity.

Lifshitz holographic renormalization from anti-de Sitter 1

Lifshitz space–times with critical exponent z = 2 can be obtained by dimensional reduction of Schrödinger space–times with critical exponent z = 0. The latter space–times are asymptotically anti-de Sitter (AdS) solutions of AdS gravity coupled to an axion–dilaton system (or even just a massless scalar field). This basic observation is used to perform holographic renormalization for four-dimensional asymptotically locally Lifshitz space–times by dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically AdS space–times coupled to an axion–dilaton system. In this setup the four-dimensional structure of the Lifshitz – Fefferman-Graham expansion and the structure of the counterterm action, including the scale anomaly, will be summarized.

From Complex to Stochastic Potential: Heavy Quarkonia in the Quark-Gluon Plasma

The in-medium physics of heavy quarkonium is an ideal proving ground for our ability to connect knowledge about the fundamental laws of physics to phenomenological predictions. One possible route to take is to attempt a description of heavy quark bound states at finite temperature through a Schrödinger equation with an instantaneous potential. Here we review recent progress in devising a comprehensive approach to define such a potential from first principles QCD and extract its, in general complex, values from non-perturbative lattice QCD simulations. Based on the theory of open quantum systems we will show how to interpret the role of the imaginary part in terms of spatial decoherence by introducing the concept of a stochastic potential. Shortcomings as well as possible paths for improvement are discussed.

Excitation and charge transfer in H-H+ collisions at 5-80 keV and application to astrophysical shocks

In astrophysical regimes where the collisional excitation of hydrogen atoms is relevant, the cross-sections for the interactions of hydrogen atoms with electrons and protons are necessary for calculating line profiles and intensities. In particular, at relative velocities exceeding ∼1000 km s−1, collisional excitation by protons dominates over that by electrons. Surprisingly, the H–H+ cross-sections at these velocities do not exist for atomic levels of n≥ 4, forcing researchers to utilize extrapolation via inaccurate scaling laws. In this study, we present a faster and improved algorithm for computing cross-sections for the H–H+ collisional system, including excitation and charge transfer to the n≥ 2 levels of the hydrogen atom. We develop a code named BDSCX which directly solves the Schrödinger equation with variable (but non-adaptive) resolution and utilizes a hybrid spatial-Fourier grid. Our novel hybrid grid reduces the number of grid points needed from ∼4000n6 (for a ‘brute force’, Cartesian grid) to ∼2000n4 and speeds up the computation by a factor of ∼50 for calculations going up to n= 4. We present (l, m)-resolved results for charge transfer and excitation final states for n= 2–4 and for projectile energies of 5–80 keV, as well as fitting functions for the cross-sections. The ability to accurately compute H–H+ cross-sections to n= 4 allows us to calculate the Balmer decrement, the ratio of Hα to Hβ line intensities. We find that the Balmer decrement starts to increase beyond its largely constant value of 2–3 below 10 keV, reaching values of 4–5 at 5 keV, thus complicating its use as a diagnostic of dust extinction when fast (∼1000 km s−1) shocks are impinging upon the ambient interstellar medium.