Control of Open Quantum Systems: A Theoretical Approach to Control of Surface Photochemistry


 Elinor Lamb
 1 years ago
 Views:
Transcription
1 Control of Open Quantum Systems: A Theoretical Approach to Control of Surface Photochemistry Von der Fakultät für Mathematik und Naturwissenschaften der Carl von Ossietzky Universität Oldenburg zur Erlangung des Grades und Titels eines Doktors der Naturwissenschaften (Dr. rer. nat.) angenommene Dissertation von Herrn Erik Asplund geboren am in Nynäshamn Oldenburg 2011
2 Erstgutachter: Zweitgutachter: Prof. Dr. T. Klüner Prof. Dr. M. Holthaus Tag der Disputation: 28. März 2011
3 To my wife
4 Erwin kann mit seinem psi kalkulieren wie noch nie. Doch wird jeder leicht einsehen, psi läßt sich nicht recht verstehen. Erich Armand Arthur Joseph Hückel
5 vii Abstract The topic of this thesis was the control of open quantum systems with the emphasis on the control of surface photochemical reactions. An open quantum system is a quantum system which interacts with its surroundings. A quantum system in condensed phase, which undergoes strong dissipative processes, is an open quantum system. The last decades have seen both experimental and theoretical approaches for gaining control over dissipative processes. From the theoretical viewpoint, it is important to model such processes in a rigorous way. In this thesis, the description of open quantum systems was realized within the Surrogate Hamiltonian approach. If systems, for which excitation and deexcitation processes have comparable timescales, are investigated, a nonmarkovian approach should be employed. The Surrogate Hamiltonian approach, which is a nonmarkovian approach, is an effective scheme, not relying on the separation of timescales, to simulate the dynamics of a system subjected to dissipation. Besides the traditional formulation of dissipative processes through the spectral density, the Surrogate Hamiltonian method enables a microscopic description of excitation and relaxation processes for open quantum systems. An efficient and accurate method to find control fields is optimal control theory (OCT). In this thesis, a control scheme relying on OCT with timedependent targets was employed to minimize the dissipative behavior of open quantum systems. Furthermore, the pulse design equations were derived and the OCT iteration algorithm was presented. To gain control of open quantum systems, the Surrogate Hamiltonian approach and OCT, with timedependent targets, were combined. Three open quantum systems were investigated by the combined method, a harmonic oscillator immersed in an ohmic bath, CO adsorbed on a platinum surface and NO adsorbed on a nickel oxide surface. The harmonic oscillator immersed in an ohmic bath served as a benchmark system to investigate the behavior of the combined Surrogate Hamiltonian /OCT approach. The two adsorbatesurface systems were chosen to investigate the applicability of the combined approach to surface photochemistry. To nullify the dissipation modeled within the Surrogate Hamiltonian approach, a freely propagating wave packet was used as a target in the OCT iteration algo
6 viii rithm, i.e. the aim of the control was to follow a predefined evolution of a wave packet. The controllability of the systems was monitored, i.e. to which degree dissipation could be surpressed. It was observed that dissipation could be subdued to a high degree for all three systems. Furthermore, the influence of the control pulses on the expectation values of the investigated systems was studied.
7 ix Kurzfassung Das Thema dieser Arbeit war die Kontrolle offener Quantensysteme, wobei der Schwerpunkt auf der Kontrolle photochemischer Oberflächenreaktionen lag. Ein offenes Quantensystem ist ein Quantensystem, das mit seiner Umgebung wechselwirkt. Ein quantenmechanisches System in kondensiertem Zustand, das starke dissipative Prozesse erfährt, ist ein offenes quantenmechanisches System. In den letzten Jahrzehnten wurden sowohl experimentelle als auch theoretische Ansätze zur die Kontrolle von dissipativen Prozessen vorgeschlagen. Um dissipative Prozesse theoretisch zu beschreiben ist eine gründliche Annäherung nötig. In dieser Arbeit wurde die Beschreibung offener Quantensysteme innerhalb der,,surrogate Hamiltonian Methode realisiert. Wenn Systeme untersucht werden, bei denen Anregungs und RelaxationsProzesse vergleichbare Zeitskalen haben, sollten nichtmarkovansätze verwendet werden. Die Methode des,,surrogate Hamiltonian, die ein nichtmarkovansatz ist, ist eine effiziente Beschreibung, die sich nicht auf die Trennung der Zeitskalen verlässt, um die Dynamik eines dissipativen Systems zu beschreiben. Neben der traditionellen Formulierung dissipativer Prozesse durch die Spektraldichte, ermöglicht die Methode des,,surrogate Hamiltonian eine mikroskopische Beschreibung der Anregungs und RelaxationsProzesse offener Quantensysteme. Eine effiziente und genaue Methode zur Berechnung von Kontrollfelder, ist die optimale Kontrolltheorie (Opimal Control Theory (OCT)). In dieser Arbeit wurde ein Kontrollkonzept, das auf der optimalen Kontrolltheorie mit zeitabhängiger Zielfunktion beruht, verwendet, um das dissipative Verhalten offener Quantensysteme zu minimieren. Außerdem wurden die Gleichungen für die Berechnung optimaler Pulse bestimmt und der iterative Algorithmus zur Berechnung optimaler Pulse vorgestellt. Um Kontrolle über offene Quantensysteme zu erzielen, wurden der,,surrogate Hamiltonian Ansatz und die optimale Kontrolltheorie mit zeitabhängigen Zielfunktionen miteinander verbunden. Drei offene Quantensysteme wurden mittels der kombinierten Methode untersucht, ein harmonischer Oszillator eingetaucht in ein ohmsches Bad, CO adsorbiert auf einer PlatinOberfläche und NO adsorbiert auf einer Nickeloxid Oberfläche. Der harmonische Oszillator, eingetaucht in einem ohmschen Bad, diente als ein BenchmarkSystem, um das Verhalten des kombinierten,,surro
8 x gate Hamiltonian /OCT Ansatzes zu untersuchen. Die beiden AdsorbatSubstrat Systeme wurden ausgewählt, um die Anwendbarkeit des kombinierten Ansatzes auf photochemische Oberflächenreaktionen zu untersuchen. Um die Dissipation, modelliert mittels der Methode des,,surrogate Hamiltonian, zu unterdrücken, wurde ein frei propagiertes Wellenpaket als Zielfunktion in dem iterativen Algorithmus verwendet, d.h. das Ziel der Kontrolle war es, einer vordefinierte Zeitentwicklung eines Wellenpakets zu folgen. Des Weiteren wurde die Kontrollierbarkeit der Systeme ermittelt, d.h. es wurde untersucht zu welchem Grad Dissipation unterdrückt werden kann. Für alle drei Systeme wurde festgestellt, dass Dissipation zu einem hohen Maß gedämpft werden kann. Darüber hinaus wurde der Einfluss der Kotrollpulse auf die Erwartungswerte der untersuchten Systeme ermittelt.
9 xi
10 xii
11 Contents 1 Introduction 1 I Theory 5 2 Quantum Dissipation Open Quantum Systems The Markov Approximation/The Quantum Markovian Semigroup Master Equation NonMarkovian Approaches The Surrogate Hamiltonian Method Dissipative Processes Modeled by the Surrogate Hamiltonian Method Numerical Implementation Optical Control of Quantum Dynamics Optimal Control Theory Formulation of the Control Problem Derivation of the Control Equations Numerical Solution of the Control Equations Discrete Implementation of the Iteration Algorithm Target Operators Coherent Control Experiments Pulse Shaping Technologies Coherent Control Experiments xiii
12 xiv CONTENTS Connecting Experiment and Theory II Applications 71 4 A Harmonic Oscillator in an Ohmic Bath 73 5 CO Adsorbed on a Platinum Surface The Model Control Laser Induced Desorption of NO from NiO(100) Experimental Findings Theoretical Calculations of the Potential Energy Surfaces Microscopic Description of the Bath and Interaction Quantum Dynamical Calculations Optimization Calculations Summary and Outlook 153 Appendices 159 A Parallelization of the Surrogate Hamiltonian 161 A.1 Parallel Computer Architectures A.1.1 SISD A.1.2 SIMD A.1.3 MISD A.1.4 MIMD A.2 Shared and Distributed Memory A.2.1 Shared Memory A.2.2 Distributed Memory A.2.3 Hybrid DistributedShared Memory A.3 Parallelization Strategy A.3.1 Domain Decomposition
13 CONTENTS xv A.3.2 Compressed Row Storage (CRS) A.3.3 Communication Algorithm B The Representation of Wave Functions and Propagation Schemes 179 B.1 Numerical Representation of Wave Functions B.1.1 The Fourier Method B.1.2 Grid Change B.1.3 Spectral Range B.2 Propagation Schemes B.2.1 The Chebychev Propagator B.2.2 The Split Operator B.3 Numerical Methods for Determining Eigenfunctions and Eigenvalues. 196 B.3.1 Imaginary Time Propagation C The Interaction Hamiltonian for NO/NiO 199 Bibliography 214 List of Figures 223 List of Tables 225
14
15 Chapter 1 Introduction Quantum mechanics is one of the most fundamental physical theories and can hence not be neglected in modern physics and chemistry. Up to date, no underlying theory has been found from which quantum probabilities, i.e. all predictions derived from quatum mechanics are of a probabilistic charactar which implies that predictions are being made about the behavior of ensambles [1], can be deduced. Quantum mechanics must hence be employed when handling problems, which include quantum probabilities. Furthermore, laser techniques have been developed so far that tailored femtosecond laser pulses can be generated in laboratories. On the femtosecond time scale, matter wave packets (particletype) can be created and their coherent evolution as a singlemolecule trajectory can be observed... It also offers new possibilities for the control of reactivity and for structural femtochemistry and femtobiology [2], i.e. light interacting with atoms and molecules is not only a source of information about the atoms and molecules studied, it can also initiate chemical reactions [3, 4]. A field of chemistry, that relies on the interaction of light with atoms and molecules, is surface photochemistry, which is important in many different applications such as molecular photo devices, photochemical vapor deposition of thin layered semiconductors, sensitive optical media, and control of photochemical reaction paths [5]. The chemical reactions, which are important in these applications occur in condensed phase. Many photochemical experiments are also conducted in condensed phase, e.g. on atoms or molecules that are adsorbed on a surface. Molecules on oxide surfaces, such as nickel oxide (NiO) and chromium oxide (Cr 2 O 3 ), have 1
16 2 CHAPTER 1. INTRODUCTION been extensively studied both theoretically and experimentally [6 23]. The surfaces in such experiments constitute an environment for the molecules studied. Due to interactions between the adsorbed molecule and the surface, energy and phase exchange emerge, which in turn perturbs or even destroys quantum coherence. The degradation of quantum coherence limits the applicability of quantum phenomena in proposed technologies, e.g. quantum computing. It is hence of importance to understand and optimally control the processes, which destroy quantum coherence. One approach to gain control would be to apply external fields. The focus of this thesis is on the interactions of quantum systems and the surrounding environment and the control of such systems. If excitation, excited state dynamics and relaxation have comparable timescales, a theoretical approach is required, which describes the system accurately and does not rely on the separation of timescales. This can be achieved within the framework of the Surrogate Hamiltonian approach [24]. In this approach, a quantum system is separated into a primary system and a bath. The interactions between the primary system and the bath are introduced by explicitly immersing the primary system into the bath, which acts as dissipative environment. A quantum system in condensed phase is affected by dissipative forces and the coherent evolution of the system is hence degraded. A perfect external field could in principle be used to protect an open quantum system from decoherence. Control of surface chemical reactions by means of external light fields is a challenging problem, both theoretically and experimentally. So far, only few approaches have been employed which typically require certain assumptions and conditions. A general approach to the control of quantum systems is provided by control theory [1, 25]. Recently, Local Control Theory (LCT) was used to protect an open quantum system from decoherence and it was proposed that Optimal Control Theory (OCT) could be used to achieve superior results [26]. OCT is a theoretical tool for the design of external fields to transfer an initial state to a specific final state. Besides the traditional finaltime control algorithms [27, 28], methods exist to handle timedependent control targets [29, 30]. Recently, a hybrid timedependent/timeindependent optimal control algorithm has been developed [31]. To this date, optimal control theory has only been applied to dissipative sys
17 3 tems in the density matrix formalism of open quantum systems [32 35] or under the adiabatic approximation [36]. The advantage of the Surrogate Hamiltonian method is that it allows a wavefunction description of open quantum systems and a microscopic modeling of the bath. In this thesis, OCT with timedependent targets is combined with the Surrogate Hamiltonian method for the first time, in order to gain control of open quantum systems. Active control of the evolution of the atoms and molecules involved in surface photochemistry experiments will shed new light in surface photochemical processes. Although many different aspects of control can be thought of, this thesis emphases on the possibility to protect open quantum systems from dissipation, which is of prime importance in quantum computing. This thesis consists of two parts. The first part will cover the theoretical tools required for the treatment and control of open quantum systems, i.e. the Surrogate Hamiltonian approach and Optimal Control Theory. In the second part three different open quantum systems will be investigated with the combined Surrogate Hamiltonian /OCT approach. Chapter 2, where different theoretical approaches to treat open quantum systems are outlined, serves as an introduction to the problem of quantum dissipation. The chapter emphases on the Surrogate Hamiltonian approach, which is a potential method for the treatment of open quantum systems. The general problem of optical control of quantum systems is introduced in chapter 3. An overview of approaches for optical control of quantum systems is presented before Optimal Control Theory is introduced, where OCT with timedependent targets is highlighted. The chapter is concluded with an overview of experimental coherent control techniques. The second part begins with chapter 4, where the harmonic oscillator serves as a benchmark system to demonstrate how the combined Surrogate Hamiltonian /OCT approach works and to which extent control can be achieved. In chapter 5 the combined Surrogate Hamiltonian /OCT approach is applied to a model system representing carbon monoxide adsorbed on a platinum surface. The degree of freedom considered in the model is the vibrational COPt mode. The system will be investigated regarding the possibility to protect a vibrationally excited system from vibrational relaxation. A Surrogate Hamiltonian treatment on an ab initio level of photodesorption dynamics is presented in chapter 6, where the laser induced desorption of
18 4 CHAPTER 1. INTRODUCTION nitrogen monoxide from a nickel oxide surface is investigated. The controllability of the system and the influence of the control pulse on the desorption observables are investigated. Chapter 7 summarizes this thesis and gives an outlook into future work in the area of control of open quantum systems. In the appendix, the numerical and computational tools necessary for all calculations are rewieved. Throughout this thesis atomic units, i.e. = m e = e = a 0 = 1, have been used unless otherwise stated.
19 Part I Theory 5
20
21 Chapter 2 Quantum Dissipation Since a perfect isolation of a quantum system is impossible [1], all realistic quantum systems are influenced by there surroundings. Since the influence of the surroundings cannot be neglected, any realistic quantum system must be treated as an open system. The problem of open quantum systems has been extensively studied since the early 60 s [37, 38], although the existence of the problem was known much earlier. Quantum fluctuations and quantum dissipation are two phenomena that naturally occur when a quantum system is influenced by its surrounding, and the theory of open quantum systems has to be applied to describe and understand these phenomena. Processes in which a quantum system loses energy and/or phase due to interactions with the surrounding environment are termed quantum dissipation. Energy relaxation does not only occur in quantum systems but is also observed in classical systems, e.g. friction, while a process, in which a system loses phase, termed dephasing, is specific for quantum systems. Examples of quantum dissipation can be found in quantum optics, nuclear magnetic resonance and condensed matter physics. In condensed matter physics dissipation plays a crucial role in areas such as surface photochemistry, photosynthesis and atoms or molecules caged in a cluster [39]. The essential idea behind the theory of open quantum systems is the separation of a global quantum system into two parts. The general approach is then to derive the dynamics of one of the subsystems from a microscopic theory by elimination of the degrees of freedom of the other subsystem. This separation is not only made due 7
22 8 CHAPTER 2. QUANTUM DISSIPATION to the fact that most realistic systems are far to complicated to be described by the fundamental laws of physics, but a microscopic description of a realistic quantum system would also result in an overwhelming amount of information. Most of this information would not contribute to the understanding of the investigated system [1]. The separation of a system into two subsystems is nontrivial, the decomposition must be made in such a way that the Hamiltonian of the system is assured to be welldefined, that the subsystem representing the environment is stable and that the interaction between the two subsystems is nonsingular [39]. The subsystem representing the surrounding environment is usually treated implicitly. The surrounding environment is normally represented by an abstract description since only its influence on the other subsystem is of interest and not the surrounding environment itself. There are two classes of commonly used environment descriptions, the environment is usually modeled by an ensamble of either two level systems [40] or harmonic oscillators [41]. The basic idea behind the use of harmonic oscillators in the description of the environment steams from a normal mode analysis together with the assumption of a weak interaction between the two subsystems [41]. There exists a wide range of different methods to treat open quantum systems and many of the methods are formally equivalent. Some of the most prominent methods are the pathintegral and projection operator formulations [42]. The following section will deal with the standard approaches and concepts of the theory of open quantum systems. The section will also serve as a prerequisite before the Surrogate Hamiltonian method, which is the method of choice for the handling of open quantum systems in this thesis, is introduced. 2.1 Open Quantum Systems As already stated, the fundamental idea of the theory of open quantum systems is to assume that the total system can be split up into two parts. The notation of composite quantum systems is hence fundamental. The Hilbert space of a composite system is the tensor product space of the Hilbert spaces of the subsystems [1], H = H (1) H (2). (2.1)
23 2.1. OPEN QUANTUM SYSTEMS 9 In the above equation H (1) denotes the Hilbert space of subsystem S (1), H (2) the Hilbert space of subsystem S (2) and H is the Hilbert space of the composite system. Equation 2.1 follows from the postulates of quantum mechanics [1]. Since it is intended to develop a formalism for open quantum systems, it is further assumed that the two subsystems are allowed to interact with each other. In the treatment of open quantum systems one of the subsystems contains one or a few active degrees of freedom, this subsystem is frequently termed primary system, denoted S. The remaining degrees of freedom are taken care of by a second subsystem usually called the environment or bath [43] and is denoted B. The Hilbert space of the compsite system is hence, according to equation 2.1, written as H = H S H B. (2.2) It is usually assumed that this composite system is a closed system, but the composite system can also be coupled to yet another system [44]. The Hamiltonian of the primary system is denoted Ĥ S and the bath Hamiltonian is denoted Ĥ B. The coupling between the primary system and the bath degrees of freedom is taken care of by an interaction Hamiltonian, which is denoted Ĥ SB. The Hamiltonian of the composite system is thus written as [1, 39] Ĥ = Ĥ S Î B + Ĥ B Î S + Ĥ SB, (2.3) where ÎS and ÎB are the identity operators of the primary system and the bath, respectively [45]. Since the evolution of an open quantum system cannot generally be represented in terms of a unitary time evolution, its dynamic is often formulated by an eligible equation of motion for its density matrix, a quantum master equation [1]. The density matrix is given by the expression [39] ˆρ = Ψ Ψ, (2.4) where Ψ is the wave function of the composite system. Since only observables of the primary system are of interest, it is convenient to introduce the reduced density matrix, ˆρ S tr Bˆρ, (2.5)
24 10 CHAPTER 2. QUANTUM DISSIPATION where ˆρ is the density matrix of the composite system and tr B denotes the partial trace over the bath degrees of freedom. The operators of the primary system are of the form Â S Î B [1], where Â S is an arbitrary operator that acts on the primary system Hilbert space, H S. The expectation value of the operator Â S is then determined by [39] Â S = tr S {Â Sˆρ S }, (2.6) where tr S denotes the partial trace over primary system s Hilbert space [1]. The density matrix of the composite system evolves unitary [45] and its time evolution is given by [1] ˆρ(t) = Û t t 0ˆρ(t 0 )(Û t t 0 ), (2.7) where Û t t 0 is the timeevolution operator, also called the propagator. The timeevolution of the reduced density matrix is hence, in accordance with equation 2.5, given by ˆρ S (t) = tr Bˆρ(t) = tr B {Ût t0ˆρ(t 0 )(Û t t 0 ) }. (2.8) By taking the time derivate of equation 2.7 and replacing the time derivative of the wavefunction with the Hamiltonian, the equation of motion for the density matrix is obtained [46], i tˆρ(t) = [Ĥ, ˆρ(t) ]. (2.9) The above equation is commonly referred to as the Liouvillevon Neumann equation. Analogically with equation 2.9 the equation of motion of the reduced density matrix is given by i tˆρ S(t) = tr B [Ĥ, ˆρ(t) ]. (2.10) In the next sections, a brief review of the most applied approaches and approximations to the above equation of motion will be given. 2.2 The Markov Approximation/The Quantum Markovian Semigroup Master Equation Assuming that there are no initial correlations between the primary system and the bath at the initial time t = t 0, makes it possible to write the initial state of the
25 2.2. MARKOV APPROXIMATION 11 composite system as ˆρ(t 0 ) = ˆρ S (t 0 ) ˆρ B, (2.11) where ˆρ B tr Sˆρ. Using the above assumption and remembering that the dynamics of the primary system is given by equation 2.8, the transformation describing the evolution of the primary system from the initial time t 0 to some later time t > t 0 may be written as ˆρ S (t) = tr B {Ût t0 (ˆρ S (t 0 ) ˆρ B ) (Û t t 0 ) }. (2.12) The dynamics expressed through the above equation can be viewed as a map, V (t fix ), which maps the initial state of the primary system, ˆρ S (t 0 ), to another state, ˆρ S (t fix ), at a later fixed time t = t fix [1], ˆρ S (t 0 ) ˆρ S (t fix ) = V (t fix )ˆρ S (t 0 ). (2.13) This map is known as a dynamical map. If t fix is allowed to vary, i.e. t fix = t, a oneparameter family of dynamical maps is provided, {V (t) t 0}, which determines the future evolution of the primary system [1]. If it is assumed that the time scale of the evolution of the primary system is much greater than the time scale of the bath correlation function, it is justified to omit memory effects for the dynamics of the primary system [1], i.e. the future of the reduced density matrix is independent of its past history. When memory effects are neglected, the process is a so called Markov process. A Markovian behavior can be formalized by the semigroup property [1]: V (t 1 )V (t 2 ) = V (t 1 + t 2 ), t 1, t 2 0, (2.14) which is a quantum dynamical semigroup. The quantum dynamical semigroup can be represented in exponential form, V (t) = e Lt, (2.15) where L is the generator of the semigroup. The equation of motion for the reduced density matrix is hence given by d dtˆρ S(t) = Lˆρ S (t). (2.16)
26 12 CHAPTER 2. QUANTUM DISSIPATION The above equation is called the Markovian quantum master equation [47] and the generator L for an Nlevel system in its most general form is given by [48] Lˆρ S (t) = i[ĥ S, ˆρ S (t)] + N 2 1 α=1 γ α (Ĝ αˆρ S (t)ĝ α 1 ) 2 {Ĝ αĝα, ˆρ S (t)}, (2.17) where Ĝ α and Ĝ α are orthonormal system operators, usually referred to as Lindblad operators, with the corresponding relaxation times γ α. Equation 2.16 is mostly called the Lindblad equation. A quantum master equation, whose generator is of the Lindblad form, can be derived for a number of physical problems using a microscopic theory with certain approximations. The low density limit, the singular coupling limit and the weakcoupling limit are some of the most important cases [45]. In some physical problems, nonlinear quantum master equations with generators of the Lindblad form are encountered. The coefficients of the Lindblad generators in such problems depend parametrically on the density matrix, which leads to a quantum master equation of the form L L(ˆρ), (2.18) d dtˆρ S(t) = L(ˆρ S (t))ˆρ S (t). (2.19) The mean field master equation and the quantum Boltzmann equation are some examples of nonlinear quantum master equations [45]. 2.3 NonMarkovian Approaches In the Markov approach it is assumed that the bath correlations decay on a much shorter timescale than the timescale of the primary system. In many physical and chemical problems this is not the case, e.g. for systems with a moderate to strong systembath coupling [39]. The starting point for most nonmarkovian approaches is, as in the Markov approximation, provided by a microscopic theory for the combined system, S +B. The first step in a theoretical treatment of nonmarkovian processes is to derive an exact equation of motion for the primary system through the elimination of the dynamical variables of the bath [45]. Analytical expressions for the evolution
27 2.3. NONMARKOVIAN APPROACHES 13 of the reduced density matrix only exist for a few simple problems, e.g. for a damped harmonic oscillator. In most cases, an exact representation of the reduced density matrix is the starting point for a perturbative treatment of the primary systembath coupling [45]. One approach to obtain the equation of motion for a nonmarkovian open system is provided by projection operator techniques. The basic idea of projection operator techniques is to regard the operation of tracing over the bath as a projection, P. The density matrix Pρ is called the relevant part of the density matrix. Hence, the aim is to derive an equation of motion for the relevant part, Pρ. The NakajimaZwanzig projection operator technique is probably the most prominent of the projection operator techniques. In this approach, a formally exact equation of motion for the primary system, which involves a certain memory kernel, is derived [45]. A disadvantage with this approach is that the perturbative expansion of the memory kernel gives rise to an integrodifferential equation, whose numerical solution can be quite demanding [1]. Another strategy is to derive a timelocal master equation for the reduced density matrix. In this approach, the equation of motion is given by d dt ρ S(t) = K(t)ρ S (t), (2.20) where K is a timedependent generator. It can be shown, employing the timeconvolutionless (TCL) projection operator technique [1], that a master equation of the above form exists for weak and intermediate coupling. Equation 2.20 is local in time and the integration over its past is hence needed [45]. A different approach to treat open systems is the path integral formulation of quantum mechanics. The state of the system, the wave function or the density matrix, at a specific time t in this approach is given by the propagator, which is the sum over all possible paths starting at a specified point at time t = 0 [39]. In contrast to the previously mentioned methods, the integration over the bath degrees of freedom gives rise to an influence functional, which contains correlations in time between different paths [49]. NonMarkovian processes can hence, in principle, be accounted for. The influence functional for a linearly coupled bath of harmonic oscillators can be derived analytically and the result is the famous CaldeiraLeggett model [50]. In spite of efforts trying to derive an influence functional for a system nonlinearly coupled to a bath of anharmonic oscillators, the strength of the path
Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationSecond postulate of Quantum mechanics: If a system is in a quantum state represented by a wavefunction ψ, then 2
. POSTULATES OF QUANTUM MECHANICS. Introducing the state function Quantum physicists are interested in all kinds of physical systems (photons, conduction electrons in metals and semiconductors, atoms,
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationNMR SPECTROSCOPY. Basic Principles, Concepts, and Applications in Chemistry. Harald Günther University of Siegen, Siegen, Germany.
NMR SPECTROSCOPY Basic Principles, Concepts, and Applications in Chemistry Harald Günther University of Siegen, Siegen, Germany Second Edition Translated by Harald Günther JOHN WILEY & SONS Chichester
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for nonphysics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationNMR and IR spectra & vibrational analysis
Lab 5: NMR and IR spectra & vibrational analysis A brief theoretical background 1 Some of the available chemical quantum methods for calculating NMR chemical shifts are based on the HartreeFock selfconsistent
More information thus, the total number of atoms per second that absorb a photon is
Stimulated Emission of Radiation  stimulated emission is referring to the emission of radiation (a photon) from one quantum system at its transition frequency induced by the presence of other photons
More informationThe Role of Electric Polarization in Nonlinear optics
The Role of Electric Polarization in Nonlinear optics Sumith Doluweera Department of Physics University of Cincinnati Cincinnati, Ohio 45221 Abstract Nonlinear optics became a very active field of research
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, AppI Li. 7 1 4 Ga. 4 7, 6 1,2
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationCHEM6085: Density Functional Theory Lecture 2. Hamiltonian operators for molecules
CHEM6085: Density Functional Theory Lecture 2 Hamiltonian operators for molecules C.K. Skylaris 1 The (timeindependent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction
More informationDO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS
DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS Quantum Mechanics or wave mechanics is the best mathematical theory used today to describe and predict the behaviour of particles and waves.
More informationNMR for Physical and Biological Scientists Thomas C. Pochapsky and Susan Sondej Pochapsky Table of Contents
Preface Symbols and fundamental constants 1. What is spectroscopy? A semiclassical description of spectroscopy Damped harmonics Quantum oscillators The spectroscopic experiment Ensembles and coherence
More informationfor High Performance Computing
Technische Universität München Institut für Informatik Lehrstuhl für Rechnertechnik und Rechnerorganisation Automatic Performance Engineering Workflows for High Performance Computing Ventsislav Petkov
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationENERGY TRANSFER IN THE WEAK AND STRONG COUPLING REGIME
ERC Starting Grant 2011 Dipar)mento di Scienze Chimiche Università degli Studi di Padova via Marzolo 1, 35131 Padova Italy ENERGY TRANSFER IN THE WEAK AND STRONG COUPLING REGIME [1] Vekshin, N. L. Energy
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
More informationThe Metropolis Algorithm
The Metropolis Algorithm Statistical Systems and Simulated Annealing Physics 170 When studying systems with a great many particles, it becomes clear that the number of possible configurations becomes exceedingly
More informationIndiana's Academic Standards 2010 ICP Indiana's Academic Standards 2016 ICP. map) that describe the relationship acceleration, velocity and distance.
.1.1 Measure the motion of objects to understand.1.1 Develop graphical, the relationships among distance, velocity and mathematical, and pictorial acceleration. Develop deeper understanding through representations
More informationAMPLIFICATION OF ATOMIC WAVES BY STIMULATED EMISSION OF ATOMS. Christian J. Borde
AMPLIFIATION OF ATOMI WAVES BY STIMULATED EMISSION OF ATOMS hristian J. Borde Laboratoire de Physique des Lasers, NRS/URA 8, Universite ParisNord, Villetaneuse, France. INTRODUTION: The recent development
More informationChapter 9 Summary and outlook
Chapter 9 Summary and outlook This thesis aimed to address two problems of plasma astrophysics: how are cosmic plasmas isotropized (A 1), and why does the equipartition of the magnetic field energy density
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information particle with kinetic energy E strikes a barrier with height U 0 > E and width L.  classically the particle cannot overcome the barrier
Tunnel Effect:  particle with kinetic energy E strikes a barrier with height U 0 > E and width L  classically the particle cannot overcome the barrier  quantum mechanically the particle can penetrated
More informationPrerequisite: High School Chemistry.
ACT 101 Financial Accounting The course will provide the student with a fundamental understanding of accounting as a means for decision making by integrating preparation of financial information and written
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationPHYS 1624 University Physics I. PHYS 2644 University Physics II
PHYS 1624 Physics I An introduction to mechanics, heat, and wave motion. This is a calculus based course for Scientists and Engineers. 4 hours (3 lecture/3 lab) Prerequisites: Credit for MATH 2413 (Calculus
More informationPotential Energy Surfaces C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology
Potential Energy Surfaces C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology Potential Energy Surfaces A potential energy surface is a mathematical function that gives
More informationAn Introduction to HartreeFock Molecular Orbital Theory
An Introduction to HartreeFock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction HartreeFock theory is fundamental
More informationCarbon Dioxide and an Argon + Nitrogen Mixture. Measurement of C p /C v for Argon, Nitrogen, Stephen Lucas 05/11/10
Carbon Dioxide and an Argon + Nitrogen Mixture Measurement of C p /C v for Argon, Nitrogen, Stephen Lucas 05/11/10 Measurement of C p /C v for Argon, Nitrogen, Carbon Dioxide and an Argon + Nitrogen Mixture
More informationThe Physics Degree. Graduate Skills Base and the Core of Physics
The Physics Degree Graduate Skills Base and the Core of Physics Version date: September 2011 THE PHYSICS DEGREE This document details the skills and achievements that graduates of accredited degree programmes
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3. 1 Basics: equations of continuum mechanics  balance equations for mass and momentum  balance equations for the energy and the chemical
More informationThe Schrödinger Equation
The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationQuantum Mechanics: Postulates
Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, xray, etc.) through
More informationBoltzmann Distribution Law
Boltzmann Distribution Law The motion of molecules is extremely chaotic Any individual molecule is colliding with others at an enormous rate Typically at a rate of a billion times per second We introduce
More informationBasic Concepts of Thermodynamics
Basic Concepts of Thermodynamics Every science has its own unique vocabulary associated with it. recise definition of basic concepts forms a sound foundation for development of a science and prevents possible
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationIntroduction. 1.1 Motivation. Chapter 1
Chapter 1 Introduction The automotive, aerospace and building sectors have traditionally used simulation programs to improve their products or services, focusing their computations in a few major physical
More information0.1 Phase Estimation Technique
Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design
More informationTargeted Advertising and Consumer Privacy Concerns Experimental Studies in an Internet Context
TECHNISCHE UNIVERSITAT MUNCHEN Lehrstuhl fur Betriebswirtschaftslehre  Dienstleistungsund Technologiemarketing Targeted Advertising and Consumer Privacy Concerns Experimental Studies in an Internet Context
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMETHODOLOGICAL CONSIDERATIONS OF DRIVE SYSTEM SIMULATION, WHEN COUPLING FINITE ELEMENT MACHINE MODELS WITH THE CIRCUIT SIMULATOR MODELS OF CONVERTERS.
SEDM 24 June 16th  18th, CPRI (Italy) METHODOLOGICL CONSIDERTIONS OF DRIVE SYSTEM SIMULTION, WHEN COUPLING FINITE ELEMENT MCHINE MODELS WITH THE CIRCUIT SIMULTOR MODELS OF CONVERTERS. Áron Szûcs BB Electrical
More informationHeating & Cooling in Molecular Clouds
Lecture 8: Cloud Stability Heating & Cooling in Molecular Clouds Balance of heating and cooling processes helps to set the temperature in the gas. This then sets the minimum internal pressure in a core
More information= N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0
Chapter 1 ThomasFermi Theory The ThomasFermi theory provides a functional form for the kinetic energy of a noninteracting electron gas in some known external potential V (r) (usually due to impurities)
More informationBerufsakademie Mannheim University of Cooperative Education Department of Information Technology (International)
Berufsakademie Mannheim University of Cooperative Education Department of Information Technology (International) Guidelines for the Conduct of Independent (Research) Projects 5th/6th Semester 1.) Objective:
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationVARIANCE REDUCTION TECHNIQUES FOR IMPLICIT MONTE CARLO SIMULATIONS
VARIANCE REDUCTION TECHNIQUES FOR IMPLICIT MONTE CARLO SIMULATIONS An Undergraduate Research Scholars Thesis by JACOB TAYLOR LANDMAN Submitted to Honors and Undergraduate Research Texas A&M University
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationWhat is molecular dynamics (MD) simulation and how does it work?
What is molecular dynamics (MD) simulation and how does it work? A lecture for CHM425/525 Fall 2011 The underlying physical laws necessary for the mathematical theory of a large part of physics and the
More informationLecture 2: Essential quantum mechanics
Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics JaniPetri Martikainen
More informationInfrared Spectroscopy: Theory
u Chapter 15 Infrared Spectroscopy: Theory An important tool of the organic chemist is Infrared Spectroscopy, or IR. IR spectra are acquired on a special instrument, called an IR spectrometer. IR is used
More informationExcitation transfer and energy exchange processes for modeling the FleischmannPons excess heat effect
Hagelstein, P.L. and I. Chaudhary. Excitation transfer and energy exchange processes for modeling the FleischmannPons excess heat effect. in ICCF14 International Conference on Condensed Matter Nuclear
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More information"in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". h is the Planck constant he called it
1 2 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". h is the Planck constant he called it the quantum of action 3 Newton believed in the corpuscular
More informationPart IV. Conclusions
Part IV Conclusions 189 Chapter 9 Conclusions and Future Work CFD studies of premixed laminar and turbulent combustion dynamics have been conducted. These studies were aimed at explaining physical phenomena
More information1 Variational calculation of a 1D bound state
TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,
More informationModule 1: Quantum Mechanics  2
Quantum Mechanics  Assignment Question: Module 1 Quantum Mechanics Module 1: Quantum Mechanics  01. (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization
More informationCLASSICAL CONCEPT REVIEW 8
CLASSICAL CONCEPT REVIEW 8 Kinetic Theory Information concerning the initial motions of each of the atoms of macroscopic systems is not accessible, nor do we have the computational capability even with
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infrared spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationMultiChannel Distribution Strategies in the Financial Services Industry
MultiChannel Distribution Strategies in the Financial Services Industry DISSERTATION der Universität St. Gallen, Hochschule für Wirtschafts, Rechts und Sozialwissenschaften (HSG) zur Erlangung der Würde
More informationNDSU Department of Physics. Graduate Student Handbook
NDSU Department of Physics Graduate Student Handbook Department of Physics North Dakota State University Fargo, ND 581086050 History Draft: August 24, 2014 Table of Contents 1. Contact 2 2. Graduate Program
More informationQuantum Computing and Grover s Algorithm
Quantum Computing and Grover s Algorithm Matthew Hayward January 14, 2015 1 Contents 1 Motivation for Study of Quantum Computing 3 1.1 A Killer App for Quantum Computing.............. 3 2 The Quantum Computer
More informationBOX. The density operator or density matrix for the ensemble or mixture of states with probabilities is given by
2.4 Density operator/matrix Ensemble of pure states gives a mixed state BOX The density operator or density matrix for the ensemble or mixture of states with probabilities is given by Note: Once mixed,
More informationESSENTIAL COMPUTATIONAL FLUID DYNAMICS
ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References
More informationQuantum Computing. Robert Sizemore
Quantum Computing Robert Sizemore Outline Introduction: What is quantum computing? What use is quantum computing? Overview of Quantum Systems Dirac notation & wave functions Two level systems Classical
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationAuxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus
Auxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationMehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics
INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree
More informationQuantum control of individual electron and nuclear spins in diamond lattice
Quantum control of individual electron and nuclear spins in diamond lattice Mikhail Lukin Physics Department, Harvard University Collaborators: L.Childress, M.Gurudev Dutt, J.Taylor, D.Chang, L.Jiang,A.Zibrov
More informationNMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing
NMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing Alex Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu
More information2.2 Creaseness operator
2.2. Creaseness operator 31 2.2 Creaseness operator Antonio López, a member of our group, has studied for his PhD dissertation the differential operators described in this section [72]. He has compared
More informationThe Kinetic Theory of Gases Sections Covered in the Text: Chapter 18
The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and
More informationRate Equations and Detailed Balance
Rate Equations and Detailed Balance Initial question: Last time we mentioned astrophysical masers. Why can they exist spontaneously? Could there be astrophysical lasers, i.e., ones that emit in the optical?
More informationIn a cyclic transformation, where the final state of a system is the same as the initial one, U = 0
Chapter 4 Entropy and second law of thermodynamics 4.1 Carnot cycle In a cyclic transformation, where the final state of a system is the same as the initial one, U = 0 since the internal energy U is a
More informationEuropean Benchmark for Physics Bachelor Degree
European Benchmark for Physics Bachelor Degree 1. Summary This is a proposal to produce a common European Benchmark framework for Bachelor degrees in Physics. The purpose is to help implement the common
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More information3D WAVEGUIDE MODELING AND SIMULATION USING SBFEM
3D WAVEGUIDE MODELING AND SIMULATION USING SBFEM Fabian Krome, Hauke Gravenkamp BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin, Germany email: Fabian.Krome@BAM.de
More informationINTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS
INTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,
More informationDISTANCE DEGREE PROGRAM CURRICULUM NOTE:
Bachelor of Science in Electrical Engineering DISTANCE DEGREE PROGRAM CURRICULUM NOTE: Some Courses May Not Be Offered At A Distance Every Semester. Chem 121C General Chemistry I 3 Credits Online Fall
More information1 The water molecule and hydrogen bonds in water
The Physics and Chemistry of Water 1 The water molecule and hydrogen bonds in water Stoichiometric composition H 2 O the average lifetime of a molecule is 1 ms due to proton exchange (catalysed by acids
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationQuantum Monte Carlo and the negative sign problem
Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich UweJens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:
More informationConcept 2. A. Description of lightmatter interaction B. Quantitatities in spectroscopy
Concept 2 A. Description of lightmatter interaction B. Quantitatities in spectroscopy Dipole approximation Rabi oscillations Einstein kinetics in twolevel system B. Absorption: quantitative description
More informationMASTER OF SCIENCE IN MECHANICAL ENGINEERING
MASTER OF SCIENCE IN MECHANICAL ENGINEERING Introduction There are over 22 schools in Mindanao that offer Bachelor of Science in Mechanical Engineering and majority of their faculty members do not have
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Javier Enciso encisomo@in.tum.de Joint Advanced Student School 009 Technische Universität München April, 009 Abstract In this paper, a gentle introduction to Quantum Computing
More informationCustomer Intimacy Analytics
Customer Intimacy Analytics Leveraging Operational Data to Assess Customer Knowledge and Relationships and to Measure their Business Impact by Francois Habryn Scientific Publishing CUSTOMER INTIMACY ANALYTICS
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationThermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearestneighbor interactions
Thermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearestneighbor interactions D. L. Huber Department of Physics, University of WisconsinMadison, Madison, WI 53706 Abstract The purpose
More information