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Concept

Momentum operator

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: \hat{p} = - i \hbar \frac{\partial}{\partial x} where ħ is Planck's reduced constant, i the imaginary unit, x is the spatial coordinate, and a partial derivative (denoted by \partial/\partial x) is used instead of a total derivative (d/dx) since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows: \hat{p}\psi = - i \hbar \frac{\partial \psi}{\partial x} In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by p, i.

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Analysis of Femtosecond Electron Bunches at the SwissFEL Injector Test Facility

Bennie Smit

The SwissFEL (Swiss Free Electron Laser) is a free electron laser which is being built at the Paul Scherrer Institute. The SwissFEL will be a user facility to study processes on unexplored scales of space and time. In order to achieve these aims, X-ray pulses on the order of femtoseconds must be generated. For such short X-ray pulses, the electron bunch length must be comparable to the final X-ray pulse length. In order to achieve such a short electron bunch length, it must be monitored at every compression stage. This work proposes two novel methods to measure such electron bunches, namely a novel spectral method that analyses the frequency content of electron bunches that becomes easier with shorter bunch length and a transverse deflecting cavity method which can conceptually measure a zero bunch length. Because the SwissFEL was not built at the time of the PhD, all experiments were performed at the SwissFEL Injector Test Facility (SITF). The SITF is a 200 MeV test bed linear electron accelerator. Theoretically the bunch length is limited due to the uncorrelated energy spread and the higher order components of the bunch compressor. However, due to the limited electron energy in combination with the short electron bunches, space charge effects dominate the beam in the transverse phase space as well as in the longitudinal phase space. A transverse deflecting cavity (TDC) measurement was used to measure a reference bunch length to which the spectral fluctuations could be correlated. In a TDC the electric fields are oriented perpendicular to the beam direction and streaks the head and tail in opposite directions onto a screen. In a conventional TDC measurement, the projected beam size onto the temporal axis is the current profile, provided that the intrinsic beam size is small. However, in the SITF the intrinsic beam size was large due to the aforementioned space charge effects. This led to the development of a novel bunch length analysis method in a space charge dominated beam. In the proposed analysis method, a dispersive section in combination with a TDC was used to visualise the longitudinal phase space. The longitudinal phase space was sliced in the momentum domain to reveal the difference between the intrinsic and the streaked beam in a momentum slice. The bunch length could be extracted by recording the times for all slices. Both the simulations and experiments suggest the method enables the measurement of bunch length in a space charge dominated beam in which the transverse intrinsic beam size is large. The measured minimal bunch length from simulations and experiments are of the same order, 19.5 fs to 24.5 fs, respectively. Since the TDC at PSI has a specified resolution of 20 fs at 200 pC, this method may be used to reduce the resolution. The spectral content of femtosecond electron bunches contains electron bunch length information. As an electron beam collides with an optical transition radiation (OTR) foil, radiation is emitted. The radiation is collected by means of a mirror setup into an optical fibre and into an optical spectrometer. The spectral range of the spectrometer is 316 THz (950 nm) to 1364 THz (220 nm). From OTR theory a flat OTR spectrum was expected and measured. As the electron bunch length was reduced below 70 fs, periodic fluctuations started to appear on the spectra. These fluctuations broaden with decreasing bunch length and show a good agreement with the bunch length measured with the TDC.

EPFL2015

Chargement

We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infinite volume. Our momentum space formulas take the form of finite sums over F-2(1) hypergeometric functions, allowing for efficient numerical evaluation. As a concrete application, we work out matrix elements for 3d phi (4)-theory, thus providing the seed ingredients for future truncation studies.

2020

Chargement

SCCER-FURIES - Best investment strategies when prosumer capacities are increased in the grid (RE Demo) - D1.4.1b

Luc Girardin, Ermanno Lo Cascio, François Maréchal, Luise Middelhauve

Political authorities, energy operators and other stakeholders have the responsibility to implement energy transition pathways by increasing decentralised renewable energy generation. As the main stakeholder, authorities often lack the appropriate tools to frame and encourage the transition and to monitor the impact of energy transition policies. On the other hand, network operators need appropriate frameworks and guidelines to implement the transition with a sustainable business perspective. This report capitalize on previous work which uses a data driven approach based on geographical information system (GIS) and machine learning techniques to generate investment pathways towards decentralised energy generation. This report takes a closer look at the economic, environmental, technical and security impact of the development of decentralized energy systems. Firstly, Chapter 1 is looking back at the definition of performance indicators, leading to a better understanding of (i) the logic behind the selection of local system boundaries to evaluate energy efficiencies, (ii) the growing importance of grey energy and (iii) the necessary shift to hourly time scale for proper life cycle assessment of the future smart grid, while on the same time, (iii) the need to consider horizon greater than 20 years for investment planning. Secondly, Chapter 2 shows how long-term investment planning methodology, which is targeting big energy consumers, can favor the emergence of future decentralized energy infrastructure, while harvesting energy from the local environment. Chapter 3 demonstrates, by optimising solar panel layout in the RE-Demo test case, that grid-aware district scale approaches are required to identify the best investment strategies. Finally, Chapter 4 presents upcoming opportunities for investments in the digitalization of future smart grids.

2021

Chargement

Fonction d'onde

thumb|300px|right|Illustration de la notion de fonction d'onde dans le cas d'un oscillateur harmonique. Le comportement en mécanique classique est représenté sur les images A et B et celui en mécaniqu

Spin

Le 'spin' () est, en physique quantique, une des propriétés internes des particules, au même titre que la masse ou la charge électrique. Comme d'autres observables quantiques, sa mesure donne des val

État quantique

L'état d'un système physique décrit tous les aspects de ce système, dans le but de prévoir les résultats des expériences que l'on peut réaliser. Le fait que la mécanique quantique soit non déterminist

PHYS-415: Particle physics I

Presentation of particle properties, their symmetries and interactions. Introduction to quantum electrodynamics and to the Feynman rules.

ChE-204: Introduction to transport phenomena

This course aims at understanding the basic equations behind macroscopic and microscopic transport phenomena (mass, heat and momentum).

PHYS-753: Dynamics of astrophysical fluids and plasmas

The dynamics of ordinary matter in the Universe follows the laws of (magneto)hydrodynamics. In this course, the system of equations that describes astrophysical fluids will be discussed on the basis of selected astrophysical examples, from the physics of stars, to galaxies and the early Universe.