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Lecture

Dispersion and Schrödinger Equation Solution

In course

PHYS-216: Mathematical methods (for SPH)

Ce cours est un complément aux cours d'analyse et d'algèbre linéaire qui apporte des méthodes et des techniques mathématiques supplémentaires requises pour les cours de physique de 3e année, notamment l'électrodynamique et la mécanique quantique. ...

Description

This lecture covers the solution of the 1D homogeneous Schrödinger equation using Fourier transforms and convolution techniques. It explores the properties of Fourier transforms, the dispersion relation, and the general solution for a wave in a one-dimensional system. The instructor demonstrates the power of convolution and Fourier transforms in obtaining d'Alembert's solution, applicable to problems like the Schrödinger equation, heat equation, and Laplace equation.

Instructors (3)

Sylvain Bréchet

Sylvain Bréchet was born on October 13th, 1981 in Moudon (legal origin Epesses, VD, Switzerland). He obtained a Master of Science in physics at EPFL in 2005. He went on to Cambridge for his PhD studies in theoretical cosmology from 2005 to 2009 under the supervision of Prof. Lasenby (FRS) and Prof. Michael Hobson at the Cavendish Laboratory of the University of Cambridge. He went back to EPFL where he is since 2010 university lecturer and research scientist in the Institute of Condensed Matter Physics. He taught classical mechanics, special relativity and thermodynamics to mechanical, electrical engineering students and physics students. He is currently writing a textbook in thermodynamics.

Jonathan Graves

Prof. Jonathan P. Graves is a Senior Scientist at EPFL and Honorary Visiting Professor at the University of York, UK. He achieved first class joint honours in Electronic Engineering and Mathematics from the University of Nottingham, UK in 1996. He completed his Ph.D. in Theoretical Mechanics from the University of Nottingham, UK, three years later in 1999. During his Ph.D. he was based in the Culham theory group of the United Kingdom Atomic Energy Authority, developing kinetic descriptions of the internal kink instability, and participating in deuterium-tritium experimental analysis in the Joint European Torus. After a short time in industry, and a postdoc at Nottingham University, he took a position at the Swiss Plasma Center at EPFL, becoming a Senior Scientist in 2014, and became an Honorary Visiting Professor at the University of York, UK, in 2020. In 2015 he became a member of the EUROfusion Scientific and Technical Advisory Committee (STAC) and a member of the EUROfusion DEMO Technical Advisory Group. He is on the editorial board for the journal Plasma Physics and Controlled Fusion, and in 2020 became Scientific Secretary of the Varenna-Lausanne International Workshop in Theory of Fusion Plasmas.

Stephan Brunner

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (28)

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

Schrödinger equation

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. The equation is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function () that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).

Non-uniform discrete Fourier transform

In applied mathematics, the nonuniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations.

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