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    Quantum Mechanics Lecture Notes. Selected Chapters

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    These are extended lecture notes of the quantum mechanics course which I am teaching in the Weizmann Institute of Science graduate physics program. They cover the topics listed below. The first four chapter are posted here. Their content is detailed on the next page. The other chapters are planned to be added in the coming months. 1. Motion in External Electromagnetic Field. Gauge Fields in Quantum Mechanics. 2. Quantum Mechanics of Electromagnetic Field 3. Photon-Matter Interactions 4. Quantization of the Schr\"odinger Field (The Second Quantization) 5. Open Systems. Density Matrix 6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation 7. Mean Field Approaches for Many Body Systems -- Fermions and Boson

    Chiral active fluids: Odd viscosity, active turbulence, and directed flows of hydrodynamic microrotors

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    While the number of publications on rotating active matter has rapidly increased in recent years, studies on purely hydrodynamically interacting rotors on the microscale are still rare, especially from the perspective of particle based hydrodynamic simulations. The work presented here targets to fill this gap. By means of high-performance computer simulations, performed in a highly parallelised fashion on graphics processing units, the dynamics of ensembles of up to 70,000 rotating colloids immersed in an explicit mesoscopic solvent consisting out of up to 30 million fluid particles, are investigated. Some of the results presented in this thesis have been worked out in collaboration with experimentalists, such that the theoretical considerations developed in this thesis are supported by experiments, and vice versa. The studied system, modelled in order to resemble the essential physics of the experimentally realisable system, consists out of rotating magnetic colloidal particles, i.e., (micro-)rotors, rotating in sync to an externally applied magnetic field, where the rotors solely interact via hydrodynamic and steric interactions. Overall, the agreement between simulations and experiments is very good, proving that hydrodynamic interactions play a key role in this and related systems. While already an isolated rotating colloid is driven out of equilibrium, only collections of two or more rotors have experimentally shown to be able to convert the rotational energy input into translational dynamics in an orbital rotating fashion. The rotating colloids inject circular flows into the fluid, such that detailed balance is broken, and it is not a priori known whether equilibrium properties of colloids can be extended to isolated rotating colloids. A joint theoretical and experimental analysis of isolated, pairs, and small groups of hydrodynamically interacting rotors is given in chapter 2. While the translational dynamics of isolated rotors effectively resemble the dynamics of non-rotating colloids, the orbital rotation of pairs of rotors can be described with leading order hydrodynamics and a two-dimensional analogy of Faxén’s law is derived. In chapter 3, a homogeneously distributed ensemble of rotors (bulk) as a realisation of a chiral active fluid is studied and it is explicitly shown computationally and experimentally that it carries odd viscosity. The mutual orbital translation of rotors and an increase of the effective solvent viscosity with rotor density lead to a non-monotonous behaviour of the average translational velocity. Meanwhile, the rotor suspension bears a finite osmotic compressibility resulting from the long-ranged nature of hydrody- namic interactions such that rotational and odd stresses are transmitted through the solvent also at small and intermediate rotor densities. Consequently, density inhomogeneities predicted for chiral active fluids with odd viscosity can be found and allow for an explicit measurement of odd viscosity in simulations and experiments. At intermediate densities, the collective dynamics shows the emergence of multi-scale vortices and chaotic motion which is identified as active turbulence with a self-similar power-law decay in the energy spectrum, showing that the injected energy on the rotor scale is transported to larger scales, similar to the inverse energy cascade of clas- sical two-dimensional turbulence. While either odd viscosity or active turbulence have been reported in chiral active matter previously, the system studied here shows that the emergence of both simultaneously is possible resulting from the osmotic compressibility and hydrodynamic mediation of odd and active stresses. The collective dynamics of colloids rotating out of phase, i.e., where a constant torque instead of a constant angular velocity is applied, is shown to be qualitatively very similar. However, at smaller densities, local density inhomogeneities imply position dependent angular velocities of the rotors resulting from inter-rotor friction. While the friction of a quasi-2D layer of active colloids with the substrate is often not easily modifiable in experiments, the incorporation of substrate friction into the simulation models typically implies a considerable increase in computational effort. In chapter 4, a very efficient way of incorporating the friction with a substrate into a two-dimensional multiparticle collision dynamics solvent is introduced, allowing for an explicit investigation of the influences of substrate on active dynamics. For the rotor fluid, it is explicitly shown that the influence of the substrate friction results in a cutoff of the hydrodynamic interaction length, such that the maximum size of the formed vortices is controlled by the substrate friction, also resulting in a cutoff in the energy spectrum, because energy is taken out of the system at the respective length. These findings are in agreement with the experiments. Since active particles in confinement are known to organise in states of collective dynamics, ensembles of rotationally actuated colloids are studied in circular confinement and in the presence of periodic obstacle lattices in chapters 5 and 6, respectively. The results show that the chaotic active turbulent transport of rotors in suspension can be enhanced and guided resulting from edge flows generated at the boundaries, as has recently been reported for a related chiral active system. The consequent collective rotor dynamics can be regarded as a superposition of active turbulent and imposed flows, leading to on average stationary flows. In contrast to the bulk dynamics, the imposed flows inject additional energy into the system on the long length scales, and the same scaling behaviour of the energy spectrum as in bulk is only obtained if the energy injection scales, due to the mutual generation of rotor translational dynamics throughout the system and the edge flows, are well separated. The combination of edge flow and entropic layering at the boundaries leads to oscillating hydrodynamic stresses and consequently to an oscillating vorticity profile. In the presence of odd viscosity, this consequently leads to non-trivial steady-state density modulations at the boundary, resulting from a balance of osmotic pressure and odd stresses. Relevant for the efficient dispersion and mixing of inert particles on the mesoscale by means of active turbulent mixing powered by rotors, a study of the dynamics of a binary mixture consisting out of rotors and passive particles is presented in chapter 7. Because the rotors are not self-propelled, but the translational dynamics is induced by the surrounding rotors, the passive particles, which do not inject further energy into the system, are transported according to the same mechanism as the rotors. The collective dynamics thus resembles the pure rotor bulk dynamics at the respective density of only rotors. However, since no odd stresses act between the passive particles, only mutual rotor interactions lead to odd stresses leading to the accumulation of rotors in the regions of positive vorticity. This density increase is associated with a pressure increase, which balances the odd stresses acting on the rotors. However, the passive particles are only subject to the accumulation induced pressure increase such that these particles are transported into the areas of low rotor concentration, i.e., the regions of negative vorticity. Under conditions of sustained vortex flow, this results in segregation of both particle types. Since local symmetry breaking can convert injected rotational into translational energy, microswimmers can be constructed out of rotor materials when a suitable breaking of symmetry is kept in the vicinity of a rotor. One hypothetical realisation, i.e., a coupled rotor pair consisting out of two rotors of opposite angular velocity and of fixed distance, termed a birotor, are studied in chapter 8. The birotor pumps the fluid into one direction and consequently translates into the opposite direction, and creates a flow field reminiscent of a source doublet, or sliplet flow field. Fixed in space the birotor might be an interesting realisation of a microfluidic pump. The trans- lational dynamics of a birotor can be mapped onto the active Brownian particle model for single swimmers. However, due to the hydrodynamic interactions among the rotors, the birotor ensemble dynamics do not show the emergence of stable motility induced clustering. The reason for this is the flow created by birotor in small aggregates which effectively pushes further arriving birotors away from small aggregates, which eventually are all dispersed by thermal fluctuations

    Examples of works to practice staccato technique in clarinet instrument

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    Klarnetin staccato tekniğini güçlendirme aşamaları eser çalışmalarıyla uygulanmıştır. Staccato geçişlerini hızlandıracak ritim ve nüans çalışmalarına yer verilmiştir. Çalışmanın en önemli amacı sadece staccato çalışması değil parmak-dilin eş zamanlı uyumunun hassasiyeti üzerinde de durulmasıdır. Staccato çalışmalarını daha verimli hale getirmek için eser çalışmasının içinde etüt çalışmasına da yer verilmiştir. Çalışmaların üzerinde titizlikle durulması staccato çalışmasının ilham verici etkisi ile müzikal kimliğe yeni bir boyut kazandırmıştır. Sekiz özgün eser çalışmasının her aşaması anlatılmıştır. Her aşamanın bir sonraki performans ve tekniği güçlendirmesi esas alınmıştır. Bu çalışmada staccato tekniğinin hangi alanlarda kullanıldığı, nasıl sonuçlar elde edildiği bilgisine yer verilmiştir. Notaların parmak ve dil uyumu ile nasıl şekilleneceği ve nasıl bir çalışma disiplini içinde gerçekleşeceği planlanmıştır. Kamış-nota-diyafram-parmak-dil-nüans ve disiplin kavramlarının staccato tekniğinde ayrılmaz bir bütün olduğu saptanmıştır. Araştırmada literatür taraması yapılarak staccato ile ilgili çalışmalar taranmıştır. Tarama sonucunda klarnet tekniğin de kullanılan staccato eser çalışmasının az olduğu tespit edilmiştir. Metot taramasında da etüt çalışmasının daha çok olduğu saptanmıştır. Böylelikle klarnetin staccato tekniğini hızlandırma ve güçlendirme çalışmaları sunulmuştur. Staccato etüt çalışmaları yapılırken, araya eser çalışmasının girmesi beyni rahatlattığı ve istekliliği daha arttırdığı gözlemlenmiştir. Staccato çalışmasını yaparken doğru bir kamış seçimi üzerinde de durulmuştur. Staccato tekniğini doğru çalışmak için doğru bir kamışın dil hızını arttırdığı saptanmıştır. Doğru bir kamış seçimi kamıştan rahat ses çıkmasına bağlıdır. Kamış, dil atma gücünü vermiyorsa daha doğru bir kamış seçiminin yapılması gerekliliği vurgulanmıştır. Staccato çalışmalarında baştan sona bir eseri yorumlamak zor olabilir. Bu açıdan çalışma, verilen müzikal nüanslara uymanın, dil atış performansını rahatlattığını ortaya koymuştur. Gelecek nesillere edinilen bilgi ve birikimlerin aktarılması ve geliştirici olması teşvik edilmiştir. Çıkacak eserlerin nasıl çözüleceği, staccato tekniğinin nasıl üstesinden gelinebileceği anlatılmıştır. Staccato tekniğinin daha kısa sürede çözüme kavuşturulması amaç edinilmiştir. Parmakların yerlerini öğrettiğimiz kadar belleğimize de çalışmaların kaydedilmesi önemlidir. Gösterilen azmin ve sabrın sonucu olarak ortaya çıkan yapıt başarıyı daha da yukarı seviyelere çıkaracaktır

    Path integrals and stochastic calculus

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    Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.Comment: 96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physic

    Quantum Integrability vs Experiments: Correlation Functions and Dynamical Structure Factors

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    Integrable Quantum Field Theories can be solved exactly using bootstrap techniques based on their elastic and factorisable S-matrix. While knowledge of the scattering amplitudes reveals the exact spectrum of particles and their on-shell dynamics, the expression of the matrix elements of the various operators allows the reconstruction of off-shell quantities such as two-point correlation functions with a high level of precision. In this review, we summarise results relevant to the contact point between theory and experiment providing a number of quantities that can be computed theoretically with great accuracy. We concentrate on universal amplitude ratios which can be determined from the measurement of generalised susceptibilities, and dynamical structure factors, which can be accessed experimentally e.g. via inelastic neutron scattering or nuclear magnetic resonance. Besides an overview of the subject and a summary of recent advances, we also present new results regarding generalised susceptibilities in the tricritical Ising universality class.Comment: 53 pages, 12 figures. arXiv admin note: text overlap with arXiv:2109.0976

    Moduli Stabilisation and the Statistics of Low-Energy Physics in the String Landscape

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    In this thesis we present a detailed analysis of the statistical properties of the type IIB flux landscape of string theory. We focus primarily on models constructed via the Large Volume Scenario (LVS) and KKLT and study the distribution of various phenomenologically relevant quantities. First, we compare our considerations with previous results and point out the importance of Kähler moduli stabilisation, which has been neglected in this context so far. We perform different moduli stabilisation procedures and compare the resulting distributions. To this end, we derive the expressions for the gravitino mass, various quantities related to axion physics and other phenomenologically interesting quantities in terms of the fundamental flux dependent quantities gsg_s, W0W_0 and n\mathfrak{n}, the parameter which specifies the nature of the non-perturbative effects. Exploiting our knowledge of the distribution of these fundamental parameters, we can derive a distribution for all the quantities we are interested in. For models that are stabilised via LVS we find a logarithmic distribution, whereas for KKLT and perturbatively stabilised models we find a power-law distribution. We continue by investigating the statistical significance of a newly found class of KKLT vacua and present a search algorithm for such constructions. We conclude by presenting an application of our findings. Given the mild preference for higher scale supersymmetry breaking, we present a model of the early universe, which allows for additional periods of early matter domination and ultimately leads to rather sharp predictions for the dark matter mass in this model. We find the dark matter mass to be in the very heavy range mχ10101011 GeVm_{\chi}\sim 10^{10}-10^{11}\text{ GeV}

    Applications of higher-form symmetries at strong and weak coupling

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    In this thesis we consider two distinct applications of higher-form symmetries in quantum field theory. First we explore the spontaneous breaking of higher-form symmetry in a holographic quantum field theory containing matter fields in the fundamental representation of the gauge group U(N). At strong coupling, we numerically solve the bulk equations of motion to compute the current-current Green’s function and demonstrate the existence of a goldstone mode. We then compare to direct analytic perturbative results obtained at weak coupling. In the second half of the thesis we work with a hydrodynamic effective field theory which possesses a higher-form symmetry. In particular, we consider a natural higher-derivative correction to force-free electrodynamics and compute a hydrodynamic transport coefficient from microscopics. Concretely, this is a perturbative QED calculation in a background magnetic field. Finally we compare our findings to astrophysical observations

    Exploring the Structure of Scattering Amplitudes in Quantum Field Theory: Scattering Equations, On-Shell Diagrams and Ambitwistor String Models in Gauge Theory and Gravity

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    In this thesis I analyse the structure of scattering amplitudes in super-symmetric gauge and gravitational theories in four dimensional spacetime, starting with a detailed review of background material accessible to a non-expert. I then analyse the 4D scattering equations, developing the theory of how they can be used to express scattering amplitudes at tree level. I go on to explain how the equations can be solved numerically using a Monte Carlo algorithm, and introduce my Mathematica package treeamps4dJAF which performs these calculations. Next I analyse the relation between the 4D scattering equations and on-shell diagrams in N = 4 super Yang-Mills, which provides a new perspective on the tree level amplitudes of the theory. I apply a similar analysis to N = 8 supergravity, developing the theory of on-shell diagrams to derive new Grassmannian integral formulae for the amplitudes of the theory. In both theories I derive a new worldsheet expression for the 4 point one loop amplitude supported on 4D scattering equations. Finally I use 4D ambitwistor string theory to analyse scattering amplitudes in N = 4 conformal supergravity, deriving new worldsheet formulae for both plane wave and non-plane wave amplitudes supported on 4D scattering equations. I introduce a new prescription to calculate the derivatives of on-shell variables with respect to momenta, and I use this to show that certain non-plane wave amplitudes can be calculated as momentum derivatives of amplitudes with plane wave states

    Growth trends and site productivity in boreal forests under management and environmental change: insights from long-term surveys and experiments in Sweden

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    Under a changing climate, current tree and stand growth information is indispensable to the carbon sink strength of boreal forests. Important questions regarding tree growth are to what extent have management and environmental change influenced it, and how it might respond in the future. In this thesis, results from five studies (Papers I-V) covering growth trends, site productivity, heterogeneity in managed forests and potentials for carbon storage in forests and harvested wood products via differing management strategies are presented. The studies were based on observations from national forest inventories and long-term experiments in Sweden. The annual height growth of Scots pine (Pinus sylvestris) and Norway spruce (Picea abies) had increased, especially after the millennium shift, while the basal area growth remains stable during the last 40 years (Papers I-II). A positive response on height growth with increasing temperature was observed. The results generally imply a changing growing condition and stand composition. In Paper III, yield capacity of conifers was analysed and compared with existing functions. The results showed that there is a bias in site productivity estimates and the new functions give better prediction of the yield capacity in Sweden. In Paper IV, the variability in stand composition was modelled as indices of heterogeneity to calibrate the relationship between basal area and leaf area index in managed stands of Norway spruce and Scots pine. The results obtained show that the stand structural heterogeneity effects here are of such a magnitude that they cannot be neglected in the implementation of hybrid growth models, especially those based on light interception and light-use efficiency. In the long-term, the net climate benefits in Swedish forests may be maximized through active forest management with high harvest levels and efficient product utilization, compared to increasing carbon storage in standing forests through land set-asides for nature conservation (Paper V). In conclusion, this thesis offers support for the development of evidence-based policy recommendations for site-adapted and sustainable management of Swedish forests in a changing climate
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