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The Bloch-Messiah theorem and its application to overlaps

Selisko, Frieder LU (2023) FYSK02 20222
Mathematical Physics
Department of Physics
Abstract
The Bogoliubov transformation is the starting point for the Hartree-Fock-Bogoliubov method and thus for many theoretical approaches that describe heavy nuclei. This transformation is based on two matrices, denoted by U and V, which are called Bogoliubov amplitudes. In this paper the properties of these matrices are investigated. Bogoliubov amplitudes are generated using BCS wavefunctions. The Bloch-Messiah theorem and the resulting Bloch-Messiah decomposition are discussed as well as a newfound restriction on the transformation matrix D from the BM decomposition. Different formulas are presented to compute the overlap between the generated wavefunctions.
Popular Abstract
The understanding of atoms has been a challenge since Democritus postulated his atomic theory of the universe. Through the times our understanding of this topic has been improved with the aid of more and more complex models. After the discovery of the atomic nucleus by Ernest Rutherford a new research topic, nuclear physics, opened and the importance of the atomic nucleus became evident. In the popular mind, nuclear physics is mainly connected to fusion and fission processes. However, initially the aim is to describe any type of nucleus as precisely as possible. Within current research, there are two possible approaches that allow a better understanding of atomic nuclei. On the one hand there are observations in an experimental setup and... (More)
The understanding of atoms has been a challenge since Democritus postulated his atomic theory of the universe. Through the times our understanding of this topic has been improved with the aid of more and more complex models. After the discovery of the atomic nucleus by Ernest Rutherford a new research topic, nuclear physics, opened and the importance of the atomic nucleus became evident. In the popular mind, nuclear physics is mainly connected to fusion and fission processes. However, initially the aim is to describe any type of nucleus as precisely as possible. Within current research, there are two possible approaches that allow a better understanding of atomic nuclei. On the one hand there are observations in an experimental setup and on the other hand there are computational models. Those models are the centerpiece of theoretical nuclear physics and are especially useful when describing nuclei that are difficult to observe because they are for example very short lived.

In this thesis a very simple model is introduced and certain aspects of it are discussed. A particular emphasis is on a transformation step, which is called the Bogoliubov transformation. A very useful theorem related to this transformation is the Bloch-Messiah theorem. It divides the Bogoliubov transformation into three separate transformation that are simpler and can therefore reduce the computational cost immensely.

Additionally, modern models use a trick and instead of describing a certain nucleus, they describe a whole set of slightly different nuclei and then superimpose those "basis set" descriptions to find a final description of the nucleus. This method is called Generator Coordinate Method and among other things cuts the computation time considerably. A key necessity for this method however is the calculation of overlaps, which describe how similar the "basis set" descriptions are.

In view of this, the simple model is used to calculate said overlaps and to discuss different ways of calculating them. (Less)
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author
Selisko, Frieder LU
supervisor
organization
course
FYSK02 20222
year
type
M2 - Bachelor Degree
subject
language
English
id
9109501
date added to LUP
2023-02-02 10:53:03
date last changed
2023-02-02 10:53:03
@misc{9109501,
  abstract     = {{The Bogoliubov transformation is the starting point for the Hartree-Fock-Bogoliubov method and thus for many theoretical approaches that describe heavy nuclei. This transformation is based on two matrices, denoted by U and V, which are called Bogoliubov amplitudes. In this paper the properties of these matrices are investigated. Bogoliubov amplitudes are generated using BCS wavefunctions. The Bloch-Messiah theorem and the resulting Bloch-Messiah decomposition are discussed as well as a newfound restriction on the transformation matrix D from the BM decomposition. Different formulas are presented to compute the overlap between the generated wavefunctions.}},
  author       = {{Selisko, Frieder}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{The Bloch-Messiah theorem and its application to overlaps}},
  year         = {{2023}},
}