[pt] A simulação da estrutura eletrônica de átomos e moléculas, desde do início da década de 90, tem se mostrado uma ferramenta imprescindível para o desenvolvimento de áreas estratégicas, ainda emergentes, mas fundamentais, como por exemplo, a área de Nanotecnologia. No entanto, esse tipo de simulação, ainda hoje é de grande complexidade e exige alto poder computacional. Dessa forma, torna-se fundamental a criação de métodos de simulação mais precisos e computacionalmente menos custosos. Este trabalho utiliza Algoritmos Evolutivos e métodos de Inteligência Computacional junto de algumas ferramentas desenvolvidas e estudadas tradicionalmente pela Matemática Aplicada em cálculos de estrutura eletrônica. Em particular, são construídas novas formas de aproximação de soluções para equação de Schrodinger, que contemplem os requisitos físicos necessários. Essas soluções serão nomeadas de funções de ondas evolucionárias, que neste trabalho serão tratadas como pontos em um espaço de Hilbert formado pelo fecho em L (2) da interseção das funções definidas na semi-reta, contínuas e de derivadas contínuas, simétricas ou antissimétricas com relação à permutação de suas coordenadas. São demonstrados alguns resultados, requisitos para utilização de Algoritmos Evolucionários e Séries de Fourier Generalizadas, baseadas em polinômios de Laguerre modificados e Wavelets. Esta pesquisa é desenvolvida inicialmente para sistemas de dois elétrons, e mais tarde é estendida para sistemas mais complexos, a fim de criar uma abordagem alternativa as tradicionais. / [en] The simulation of the electronic structure of atoms and molecules has been shown to be, from the beginning of 90 s, an indispensable tool for the development of strategic areas, that are still emergent, but fundamental, like nanotechnology. However, this type of simulation is still of great complexity today and demands high computational power. Thus, the creation of more precise and less costly methods becomes fundamental. With the elaboration of this research, the intention is to create alternatives basis that can be used into the traditional methods of simulation of electronic structure, such as the Hartree-Fock method, GVB, among others. This essay intends to investigate part of the mathematical tools used in the calculations of electronic structure, in order to create disruptive approaches, related to the precision or velocity of the obtainment of relevant results. The new methods are based in Computational Intelligence and concepts of Functional Analysis like Wavelets.

Des études antérieures ont mis en évidence que le transfert de solutés neutres à travers des membranes est influencé par la présence d'ions en solution. Ainsi, la connaissance des interactions multiples à l'échelle nanométrique, entre le polymère, l'eau et les solutés (ions, espèces organiques) constituent un verrou pour l'amélioration des performances des procédés membranaires. Dans cette étude une approche multi-échelle fondamentale est proposée, combinant des outils théoriques et expérimentaux, afin d'obtenir les paramètres microscopiques et macroscopiques caractérisant les interactions étudiées pour différentes compositions ioniques. Plus précisément, il s'agit de comprendre comment les ions affectent le transfert d'un soluté organique. Dans un premier temps, certaines propriétés caractérisant l'hydratation des ions sont calculées et comparées aux flux de diffusions de sucres à travers des membranes de Nanofiltration et échangeuses d'ions obtenus pour différentes compositions ioniques. Dans un deuxième temps, des systèmes constitués d'une membrane échangeuse de cations (CMX) équilibrée avec différents cations ainsi que le glucose hydraté sont modélisés en utilisant une approche combinée Mécanique Quantique/ Mécanique Moléculaire. Cette approche a permis d'étudier la solubilité du sucre dans la matrice polymère ainsi que les interactions polymère-polymère comme l'énergie de cohésion. Enfin, l'influence des ions sur les caractéristiques physiques de la membrane CMX est étudiée en utilisant diverses méthodes expérimentales comme la détermination des angles de contacts et des spectres IR ou la mesure de la température de solidification par DSC. Les propriétés vibrationnelles sont également calculées dans le cadre de la théorie de la fonctionnelle de la densité (DFT). L'ensemble de ces données sont comparées avec les grandeurs de transport afin de valider les mécanismes moléculaires proposés. Ce travail montre que la nature des contre-ions de la membrane modifie l'énergie de cohésion entre les fragments de la membrane. Ainsi, l'énergie de cohésion influe sur la diffusion des composés organiques neutres à travers les membranes. / Previous works have shown that the transfer of neutral solutes through membranes is influenced by the presence of ions in solution. In the framework of process intensification, the knowledge of the molecular mechanisms involved is of fundamental importance to increase and predict the process performances. The aim of this Thesis is to use a combined quantum/molecular computational approach and experimental methodologies to better understand how ions can affect the solute flux. In the first part of the work, some properties of ions in solution are computed and compared with sugar fluxes through membranes for nanofiltration and electrodialysis. In the following, systems composed of Cation-exchange membrane equilibrated by different counter-ion and hydrated glucose are examined by Quantum Mechanics/Molecular Mechanics. This is done mainly to investigate the sugar solubility in the polymer matrix and diffusion related interactions like polymer chain-chain cohesion energy. In the last part, contact angle, differential scanning calorimetry and Infra-Red spectra are measured to characterize the physical properties of the membrane and possible influence of the counter-ion on cation exchange membrane. This work shows that the nature of the counter-ions modifies the cohesion energy between the membrane polymer fragments. In its turn, the cohesion energy affects the diffusion of neutral organic compounds through the membranes.

Cette thèse traite la dynamique hors équilibre des systèmes quantiques ouverts couplés à un réservoir externe. Un modèle spécifique exactement soluble, le modèle sphérique, sert comme exemple paradigmatique. Ce modèle se résout exactement en toute dimension spatiale et pour des interactions très générales. Malgré sa simplicité technique, ce modèle est intéressant car ni son comportement critique d’équilibre ni celui hors équilibre est du genre champ moyen. La présentation débute avec une revue sur la mécanique statistique des transitions de phases classique et quantique, et sur les propriétés du modèle sphérique. Sa dynamique quantique ne se décrit point à l’aide d’une équation de Langevin phénoménologique. Une description plus complète à l’aide de la théorie de l’équation de Lindblad est nécessaire. Les équations de Lindblad décrivent la relaxation d’un système quantique vers son état d’équilibre. En tant que premier exemple, le diagramme de phases dynamique d’un seul spin sphérique quantique est étudié. Réinterprétant cette solution en tant qu’une approximation champ moyen d’un problème de N corps, le diagramme de phases quantique est établi et un effet « congeler en réchauffant » quantique est démontré. Ensuite, le formalisme de Lindblad est généralisé au modèle sphérique quantique de N particules: primo, la forme précise de l’équation de Lindblad est obtenue des conditions que (i) l’état quantique d’équilibre exacte est une solution stationnaire de l’équation de Lindblad et (ii) dans le limite classique, l’équation Langevin de mouvement est retrouvée. Secundo, le modèle sphérique permet la réduction exacte du problème de N particules à une seule équation intégro-différentielle pour le paramètre sphérique. Tertio, en résolvant pour le comportement asymptotique des temps longs de cette équation, nous démontrons que dans la limite semi-classique, la dynamique quantique effective redevient équivalente à une dynamique classique, à une renormalisation quantique de la température T près. Quarto, pour une trempe quantique profonde dans la phase ordonnée, nous démontrons que la dynamique quantique dépend d’une manière non triviale de la dimension spatiale. L’émergence du comportement d’échelle dynamique et des corrections logarithmiques est discutée en détail. Les outils mathématiques de cette analyse sont des nouveaux résultats sur le comportement asymptotique de certaines fonctions hypergéométriques confluentes en deux variables / This study deals with the dynamic properties of open quantum systems far from equilibrium in d dimensions. The focus is on a special, exactly solvable model, the spherical model (SM), which is technically simple. The analysis is of interest, since the critical behaviour in and far from equilibrium not of mean-field type. We begin with a résumé of the statistical mechanics of phase transitions and treat especially the quantum version of the SM. The quantum dynamics (QD) of the model cannot be described by phenomenological Langevin equation and must be formulated with Lindblad equations.First we examine the dynamic phase diagram of a single spherical quantum spin and interpret the solution as a mean-field approximation of the N-body problem. Hereby, we find a quantum mechanical ‘freezing by heating’ effect. After that, we extend the formalism to the N-body problem, determining first the form of the Lindblad equation from consistency conditions. The SM then allows the reduction to a single integro-differential equation whose asymptotic solution shows, that the effective QD in the semi-classical limit is fully classical. For a deep quench in the ordered phase, we show that the QD strongly and non-trivially depends on d and derive the dynamic scaling behaviour and its corrections. The mathematical tools for this analysis are new results on the asymptotic behaviour of certain confluent hypergeometric functions in two variables

A presente dissertação tem como objeto de estudo sistemas da física da matéria condensada que sejam capazes de simular sistemas gravitacionais, tais como buracos negros e universos em expansão, onde processos quânticos tomam parte. Neste estudo nos debruçamos principalmente sobre o modelo do fluido e condensados de Bose-Einstein. No modelo do fluido exploramos a geometria efetiva que surge e os problemas de back-reaction e dos modos trans-planckianos de campos quânticos. No modelo baseado em condensados exploramos sua faceta cosmológica e a possibilidade de campos maciços. Além destes dois modelos de grande relevância na literatura, ainda expomos os análogos em cordas elásticas e os baseados em ondas na superfícies de fluidos e uma análise geral baseada no formalismo lagrangeano para campos. / This dissertation has as object of study systems of condensate-matter physics which can simulate gravitational systems like black holes and expanding universes where quantum processes take place. In this study we lay attention mainly on the fluid model and on Bose-Einstein-condensate-based models. In the fluid model we explore the features of the emergent geometry and other problems like the back-reaction and the trans-planckian modes of quantum fields. In the condensate-based models we explore their cosmological aspects and the possibility for massive fields. Moreover, we shall present two other models, the elastic string and the surface-wave-based models in fluids, and a very general analysis based on the Lagrangean formalism for fields.

Using the method of the star product, the non-relativistic regime of the Dirac equation is evaluated and the NC hamiltonian to the Zeeman e ect is determined. Using the rst-order perturbation theory, the correction to the energy is calculated. We obtain the orbital and spin Land e factors. It is shown that the experimental value for the spin Land e factor put the following upper limit on the magnitude of the momentum NC parameter, p . 0; 34 eV=c. Established also a possible correction of the NC phase space to the presently accepted value of Planck's constant with an uncertainty of 2 part in 1035. By mapping via Boop's shift we obtain the Landau levels and the Hall conductivity for graphene in NC phase space. Using the current experimental precision, respectively, of the Hall conductivity and of the Landau levels in graphene, we obtain the following upper limit to the magnitude of the momentum NC parameter, p . 2; 5eV=c e p . 8; 5eV=c. Finally, by Newton's law in NC space and using the Langevin equations, we describe the Browniano motion, and thus we de ne a new physical parameter that shows the possibility of detecting NC eff ects on the macroscopic scale. / Utilizando-se do m étodo do produto estrela, o regime não relativí stico da equa ção de Dirac é avaliado e o hamiltoniano NC para o efeito Zeeman é determinado. Usando a teoria de perturba ção de primeira ordem, a corre ção para a energia é calculada. Obtemos assim, os fatores de Land é orbital e de spin. É mostrado que o valor experimental para o fator de Land é de spin impõe o seguinte limite superior na magnitude do parâmetro NC de momento,p . 0; 34 eV=c. Estabelecemos tamb ém uma possí vel corre ção do espa ço de fase NC para o valor atualmente aceito da constante de Planck, com uma incerteza de 2 partes em 10 elevado a 35. Atrav és do mapeamento via Boop's shift obtemos os n íveis de Landau e a condutividade Hall para o grafeno no espa ço de fase NC. Utilizando a atual precisão experimental, respectivamente, da condutividade Hall e dos n veis de Landau no grafeno, obtemos os seguintes limites superiores para a magnitude do parâmetro NC de momento,p . 2; 5eV=c ep . 8; 5eV=c. Por fim, atrav és das leis de Newton num espa ço NC e utilizando as equa ções de Langevin, descrevemos o movimento Browniano, e assim defi nimos um novo parâmetro f ísico que mostra a possibilidade de detectar efeitos NC na escala macrosc ópica.

Path integral approaches have been widely used for long in both quantum mechanics as well as statistical mechanics. In addition to being a tool for obtaining the probability distributions of interest(wave functions in the case of quantum mechanics),these methods are very instructive and offer great insights into the problem. In this thesis, path integrals are extensively employed to study some very interesting problems in both equilibrium and non-equilibrium statistical mechanics. In the non-equilibrium regime, we have studied, using a path integral approach, a very interesting class of anomalous diffusion, viz. the L´evy flights. In equilibrium statistical mechanics, we have evaluated the partition function for a class of molecules referred to as the hindered rotors which have a barrier for internal rotation. Also, we have evaluated the exact quantum statistical mechanical propagator for a harmonic potential with a time-dependent force constant, valid under certain conditions. Diffusion processes have attracted a great amount of scientific attention because of their presence in a wide range of phenomena. Brownian motion is the most widely known class of diffusion which is usually driven by thermal noise. However ,there are other classes of diffusion which cannot be classified as Brownian motion and therefore, fall under the category of Anomalous diffusion. As the name suggests, the properties of this class of diffusion are very different from those for usual Brownian motion. We are interested in a particular class of anomalous diffusion referred to as L´evy flights in which the step sizes taken by the particle during the random walk are obtained from what is known as a L´evy distribution. The diverging mean square displacement is a very typical feature for L´evy flights as opposed to a finite mean square displacement with a linear dependence on time in the case of Brownian motion. L´evy distributions are characterized by an index α where 0 <α ≤ 2. When α =2, the distribution becomes a Gaussian and when α=1, it reduces to a Cauchy/Lorentzian distribution. In the overdamped limit of friction, the probability density or the propagator associated with L´evy flights can be described by a position space fractional Fokker-Planck equation(FFPE)[1–3]. Jespersen et al. [4]have solved the FFPE in the Fourier domain to obtain the propagator for free L´evy flight(absence of an external potential) and L´evy flights in linear and harmonic potentials. We use a path integral technique to study L´evy flights. L´evy distributions rarely have a compact analytical expression in the position space. However, their Fourier transformations are rather simple and are given by e−D │p│α where D determines the width of the distribution. Due to the absence of a simple analytical expression, attempts in the past to study L´evy flights using path integrals in the position space [5, 6] have not been very successful. In our approach, we have tried to make use of the elegant representation of the L´evy distribution in the Fourier space and therefore, we write the propagator in terms of a two-dimensional path integral –one over paths in the position space(x)and the other over paths in the Fourier space(p). We shall refer to this space as the ‘phase space’. Such a representation is similar to the Hamiltonian path integral of quantum mechanics which was introduced by Garrod[7]. If we try to perform the path integral over Fourier variables first, then what remains is the usual position space path integral for L´evy flights which is rather difficult to solve. Instead, we perform the position space path integral first which results in expressions which are rather simple to handle. Using this approach, we have obtained the propagators for free L´evy flight and L´evy flights in linear and harmonic potentials in the over damped limit [8]. The results obtained by this method are in complete agreement with those obtained by Jesepersen et al. [4]. In addition to these results, we were also able to obtain the exact propagator for L´evy flights in a harmonic potential with a time-dependent force constant which has not been reported in the literature. Another interesting problem that we have considered in the over damped limit is to obtain the probability distribution for the area under the trajectory of a L´evy particle. The distributions, again, were obtained for free L´evy flight and for L´evy flights subjected to linear and harmonic potentials. In the harmonic potential, we have considered situations where the force constant is time-dependent as well as time-independent. Like in the case of the over damped limit, the probability distribution for L´evy flights in the under damped limit of friction can also be described using a fractional Fokker-Planck equation, although in the full phase space. However, this has not yet been solved for any general value of α to obtain the complete propagator in terms of both position and velocity. Using our path integral approach, the exact full phase space propagators have been obtained for all values of α for free L´evy flights as well as in the presence of linear and harmonic potentials[8]. The results that we obtain are all exact when the potential is at the most harmonic. If the potential is higher than harmonic, like the cubic potential, we have used a semi classical evaluation where, we extremize the action using an optimal path and further, account for fluctuations around this optimal path. Such potentials are very useful in describing the problem of escape of a particle over a barrier. The barrier crossing problem is very extensively studied for Brownian motion (Kramers problem) and the associated rate constant has been calculated in a variety of methods, including the path integral approach. We are interested in its L´evy analogue where we consider the escape of a particle driven by a L´evy noise over a barrier. On extremizing the action which depends both on phase space variables, we arrived at optimal paths in both the position space as well as the space of the conjugate variable, p. The paths form an infinite hierarchy of instant on paths, all of which have to be accounted for in order to obtain the correct rate constant. Care has to be taken while accounting for fluctuations around the optimal path since these fluctuations should be independent of the time-translational mode of the instant on paths. We arrived at an ‘orthogonalization’ scheme to perform the same. Our procedure is valid in the limit when the barrier height is large(or when the diffusion constant is very small), which would ensure that there is small but a steady flux of particles over the barrier even at very large times. Unlike the traditional Kramers rate expression, the rate constant for barrier crossing assisted by L´evy noise does not have an exponential dependence on the barrier height. The rate constant for wide range of α, other than for those very close to α = 2, are proportional to Dμ where, µ ≈ 1 and D is the diffusion constant. These observations are consistent with the simulation results obtained by Chechkin et al. [9]. In addition, our approach when applied to Brownian motion, gives the correct dependence on D. In equilibrium statistical mechanics we have considered two problems. In the first one, we have evaluated the imaginary time propagator for a harmonic oscillator with a time-dependent force constant(ω2(t))exactly, when ω2(t) is of the form λ2(t) - λ˙(t)where λ(t) is any arbitrary function of t. We have made use of Hamiltonian path integrals for this. The second problem that we considered was the evaluation of the partition function for hindered rotors. Hindered rotors are molecules which have a barrier for internal rotation. The molecule behaves like free rotor when the barrier is very small in comparison with the thermal energy, and when the barrier is very high compared to thermal energy, it behaves like a harmonic oscillator. Many methods have been developed in order to obtain the partition function for a hindered rotor. However, most of them are some what ad-hoc since they interpolate between free-rotor and the harmonic oscillator limits. We have obtained the approximate partition function by writing it as the trace of the density matrix and performing a harmonic approximation around each point of the potential[10]. The density matrix for a harmonic potential is in turn obtained from a path integral approach[11]. The results that we obtain using this method are very close to the exact results for the problem obtained numerically. Also, we have devised a proper method to take the indistinguishability of particles into account in internal rotation which becomes very crucial while calculating the partition function at low temperatures.

Many quantum computation algorithms, and processes like measurement based quantum computing, require the initial state of the quantum computer to be an eigenstate of a specific unitary operator. Here we study how quantum states that are eigenstates of finite dimensional irreducible representations of the special unitary (SU(d)) and the permutation (S_n) groups can be efficiently constructed in the computational basis formed by tensor products of the qudit states. The procedure is a unitary transform, which first uses Schur-Weyl duality to map every eigenstate to a unique Schur basis state, and then recursively uses the Clebsch - Gordan transform to rotate the Schur basis state to the computational basis. We explicitly provide an efficient quantum algorithm, and the corresponding quantum logic circuit, to generate any desired eigenstate of SU(2) and SU(3) irreducible representations in the computational basis.

In this thesis, we investigate supersymmetric quantum mechanics (SUSYQM) and its relation to index theorems and equivariant cohomology. We define some basic constructions on super vector spaces in order to set the language for the rest of the thesis. The path integral in quantum mechanics is reviewed together with some related calculational methods and we give a path integral expression for the Witten index. Thereafter, we discuss the structure of SUSYQM in general. One shows that the Witten index can be taken to be the difference in dimension of the bosonic and fermionic zero energy eigenspaces. In the subsequent section, we derive index theorems. The models investigated are the supersymmetric non-linear sigma models with one or two supercharges. The former produces the index theorem for the spin-complex and the latter the Chern-Gauss-Bonnet Theorem. We then generalise to the case when a group action (by a compact connected Lie group) is included and want to consider the orbit space as the underlying space, in which case equivariant cohomology is introduced. In particular, the Weil and Cartan models are investigated and SUSYQM Lagrangians are derived using the obtained differentials. The goal was to relate this to gauge quantum mechanics, which was unfortunately not successful. However, what was shown was that the Euler characteristics of a closed oriented manifold and its homotopy quotient by U(1)n coincide.

In 2016 the Ministry of Education in British Columbia (BC), Canada introduced the topics of quantum mechanics (QM) into the Grade 8 science curriculum. Science teachers with or without QM background are expected to learn and teach QM. Stemming from a constructivist theoretical framework, this in-depth exploratory case study explores the processes of learning and teaching the topics of QM by asking: “How does a Grade 8 science teacher learn to teach QM?” The purpose was to understand the teacher’s QM learning process, the development of pedagogical content knowledge (PCK) in QM and teacher’s views of the nature of science (NOS). The data was collected through multiple sources and analyzed by using thematic analysis. The themes were identified under five main categories: 1) the development of PCK in QM is complex, 2) the student-centered approach mandated in the redesigned curriculum may be limiting, 3) the nature of learning QM is not different than learning other subjects, 4) middle school science education is inconsistent with the current level of scientific knowledge, and 5) the development of informed views of NOS requires an accumulation and synthesis of prior knowledge in history and philosophy of science (HPS). The study proposes two previously unexplored integral aspects of PCK framework, since: the ‘allotted time’ in learning and teaching a subject and ‘pre-PCK’ change the nature of PCK development. The term pre-PCK was coined referring to the specific content oriented and student-centered activities that take place before the class with the goal of establishing an effective basis for the PCK development. The insights emerging from the study would be of interest to other Grade 8 science teachers in BC, pre-service teacher program coordinators at the universities, and the Ministry of Education in BC to provide institutional support. This study would also contribute to closing the knowledge and communication gaps between the fields of science, science education practice and science education research. / Graduate

De nombreux processus biologiques liés aux membranes cellulaires lipidiques sont encore très mal connus. La présence d'eau et d'ions à l'interface influence les propriétés structurelles et dynamiques de la bicouche lipidique. Les techniques de fluorescence sont très utiles pour étudier les membranes en raison de la grande sensibilité des sondes à leur environnement. Nous avons utilisé la technique de relaxation de solvant (SR) pour explorer l'hydratation et la mobilité de l'eau. Nous avons également réalisé des calculs quantiques (QM) et des dynamiques moléculaires (DM) pour étayer nos expériences. Les résultats SR montrent qu'un petit cation (Na+) est très attiré par la membrane et augmente sa rigidité à l'opposé des cations (NH4+, Cs+) plus gros. Les anions (CI04-, SCN-) s'adsorbent à l'interface plus facilement que Cl-. Ces anions changent la mobilité et l'hydratation des têtes polaires des lipides de la bicouche. Les études SR de la zone hydrophobe de la membrane montrent que les processus de relaxation sont ici très complexes. lis reflètent des processus rapides intramoléculaire (relaxation de torsion, transferts de charge) et des processus intermoléculaires lents. Les calculs QM ont permis de créer les champs de force de trois sondes fluorescentes (Prodan, Laurdan et C-laurdan). Les simulations DM ont permis de déterminer les positions des sondes dans une membrane DOPC. La modélisation reproduit correctement les résultats SR, en particulier les temps de relaxation : de l'ordre de la ps en solvant aqueux et de la ns dans la membrane. Les simulations MD sont complémentaires des méthodes SR et permettent de surveiller le comportement de molécules uniques. / Many biologically important processes and phcnomena in lipid membranes are still not fully understood. The presence of ions and water molœules has a significant influence on the structural and dynamical properties of lipid bilayers. Fluorescent techniques are versatile tools for studying the lipid membranes, because the fluorescence emission is strongly sensitive to dye environment. We have conducted fluorescent solvent relaxation (SR) experiments to explore the hydration and mobility properties in lipid membranes in the presence of different chaotropic ions. We have also carried out Quantum Mechanical (QM) calculations and Molecular Dynamics (MD) simulations for supporting the SR experiments. SR experiments show that small cation (Na+) is attracted to the membrane and increases rigidity ofbilayer, while larger cations (NH/, Cs+) should not. Large anions (CI04·, SCN') adsorl, at the membrane interface more easily than smaller ones (Cl') and significantly change tl!e mobility and hydration of the headgroup region oflipid bilayer. SR study ofhydrophobic part of the membrane show that SR processes are complex there and reflect botl!: faster, intramolecular (torsional relaxation or fonnation of charge transfer state) and slower, intermolecular (SR) relaxation processes. QM calculatiom were used to create force-field for three fluorescent dyes (Prodan, Laurdan and C-laurdan). MD simulations allow detennining position of the dye in the lipid membrane in the ground state and after excitation and reproduce correctly SR timescale- ps in water and ns in the membrane. MD simulations extend the capabilities of SR method and allow observing the behaviour of individual molecules.