Perturbation theory

The perturbation theory is a mathematical method that examines the influence of a perturbation on an analytically solvable (known) system.

Originally it was used in connection with astronomical problems (three-body problem), but today it is mainly used in quantum theory (Lord Rayleigh, E. Schrödinger).

The basic idea is to split the Hamilton operator of the system into two parts


whereby H^0 the undisturbed (solvable) portion and λH^1 represents the disorder. If the perturbation is sufficiently small, the eigenvalues ​​and eigenfunctions can be reduced to a power seriesλto develop


which is iteratively solvable. Depending on how far the power series is developed, a distinction is made between perturbation theories of first order, second order, etc.

The most frequently used methods today are the MPn-Procedure according to Møller and Plesset.

Learning units in which the term is dealt with

Table of contents

In contrast to the standard model of particle physics, in string theory the fundamental building blocks that make up our world are not particles in the sense of points (i.e. zero-dimensional objects), but vibrating one-dimensional objects. These one-dimensional objects become Strings called (English for string). The individual elementary particles can be imagined as vibrational excitation of the strings, whereby the frequency corresponds to an energy according to quantum mechanics.

In further developments of string theory, the so-called brane theories, not only one-dimensional (or, including time, 1 + 1-dimensional) strings are regarded as basic objects, but also higher-dimensional objects (called “brane” [1]).

String theory bypasses the problems of singularities that arise in classical quantum field theory and the renormalization theory developed to tame them. They arise especially for point particles from their self-interaction, which occurs in the case of extensive z. B. one-dimensional objects is "smeared" and thus softened.

Prototype example

The earliest application of the so-called Perturbation theory consisted in dealing with the otherwise unsolvable mathematical problems of celestial mechanics: for example the orbit of the moon, which moves noticeably differently than a simple Kepler ellipse due to the competing gravity of the earth, and the sun.

Fault methods start with a simplified form of the original problem, the is easy enough to be solved exactly. In celestial mechanics this is usually a Kepler ellipse. Under Newton's gravity, an ellipse is just right when there are only two gravitating bodies (e.g. earth and moon), but not quite right when there are three or more objects (e.g. earth, moon, sun and the rest of) the solar system) and not entirely correct when the gravitational interaction is given using formulations from general relativity.

Chiral perturbation theory

I started my physics studies at Johannes Gutenberg University in the winter semester 1998/99 and after two years I finished the basic studies with the passing
of the intermediate diploma in August 2000. I then left Mainz and spent a year as an exchange student in the USA at the University of Washington in Seattle. Even before the start of my studies, I was certain that I wanted to spend part of my studies abroad, and the large number of exchange programs at Johannes Gutenberg University, especially in the physics department, was an important reason, along with the good reputation that Mainz physics enjoys my choice of university.

I returned to Mainz for the 2001/02 winter semester and continued my studies here. Since the beginning of my studies, I have been interested in theoretical physics, which I specialized in during the main course. Elementary particle and nuclear physics in particular piqued my interest. I passed the diploma exams in March 2003 and then started my diploma thesis at the Institute for Nuclear Physics.

The support from my supervisor and all other members of the group is excellent. I have the opportunity to work independently through frequent discussions with other group members, especially a fellow of the Alexander von Humboldt Foundation from Georgia, but questions that arise are answered quickly. My thesis deals with one of the fundamental interactions, the strong interaction, which among other things is cohesion
of the atomic nuclei. More precisely, I am concerned with chiral perturbation theory and its application to calculate the electromagnetic form factors of the nucleon. Chiral perturbation theory is an effective field theory and provides a description of the low energy sector of quantum chromodynamics, which is the strong interaction theory. Instead of looking at the more fundamental quarks and gluons, mesons and hadrons, especially the pions and nucleons, are the degrees of freedom of chiral perturbation theory. One of the interesting sides of my work is that it includes both formal and mathematical aspects, such as the further development of a renormalization scheme, as well as phenomenological questions, and that I can compare my results with experimental data.

Derivation / theory for the 2nd test WS 2014/15

Contribution by Nolle & raquo January 21, 2015, 8:48 am

I would be interested in what the general opinion is, what deductions could come. The fact that the Rotter absolutely had to print through the perturbation theory on Monday makes that a candidate, I think.

But otherwise?
Hydrogen atom? If so, how much how far?

I would be happy to receive suggestions as to what else you should incorporate by Friday

Re: Derivation / theory for the 2nd test WS 2014/15

Contribution by OffBeat & raquo 01/21/2015, 13:52

As I said, perturbation theory is already a hot tip!
Otherwise I would look at the eigenvalue equations, which then lead to the quantum numbers m and l (i.e. angular momentum square and z component).
What then spontaneously comes to mind would be the SGL in spherical coordinates and the transformation matrix for the two spin representations for two spins with s = 1/2.

Video: perturbation theory quantum mechanics (January 2022).