prikaz prve stranice dokumenta Jednočestični Schrödingerovi operatori
  Download
PDF 279.45 KB
master's thesis
Jednočestični Schrödingerovi operatori
Zagreb: University of Zagreb, Faculty of Science, 2021. urn:nbn:hr:217:100660
Novak, Valentino
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Novak, V. (2021). Jednočestični Schrödingerovi operatori (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:100660

Novak, Valentino. "Jednočestični Schrödingerovi operatori." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:100660

Novak, Valentino. "Jednočestični Schrödingerovi operatori." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:100660

Novak, V. (2021). 'Jednočestični Schrödingerovi operatori', Master's thesis, University of Zagreb, Faculty of Science, accessed 27 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:100660

Novak V. Jednočestični Schrödingerovi operatori [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2021 [cited 2024 November 27] Available at: https://urn.nsk.hr/urn:nbn:hr:217:100660

V. Novak, "Jednočestični Schrödingerovi operatori", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2021. Available at: https://urn.nsk.hr/urn:nbn:hr:217:100660

Metadata
TitleJednočestični Schrödingerovi operatori
Title (english)One-particle Schrödinger operators
AuthorValentino Novak
MentorMarko Erceg (mentor)
Committee memberMarko Erceg (predsjednik povjerenstva)
Committee memberMarko Vrdoljak (član povjerenstva)
Committee memberZoran Vondraček (član povjerenstva)
Committee memberMiljenko Marušić (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2021-09-24, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu dajemo pregled teorije neograničenih samoadjungiranih operatora na Hilbertovom prostoru i njihovu primjenu na modeliranje atoma vodika. Pokazujemo da glavni rezultati teorije, Spektralni teorem i Stoneov teorem vrijede isključivo za samoadjungirane operatore, pa su nam baš takvi operatori od glavnog interesa. Kako su u primjenama operatori koji obično susrećemo simetrični, pokazujemo nužne i dovoljne uvjete da bi simetričan operator bio samoadjungiran i promatramo mogućnosti proširenja simetričnih operatora do samoadjungiranih operatora. Kao glavne primjere promatramo operatore položaja i količine gibanja. U nastavku spominjemo Spektralni teorem kao jedan od glavnih rezultata funkcionalne analize, koji nam za proizvoljnu Borelovu funkciju \(f\) omogućava definiranje izraza \(f\) (A), gdje je A samoadjungirani operator. Koristeći Spektralni teorem može se pokazati Stoneov teorem koji nam dokazuje da je generator jednoparametarske unitarne grupe koja opisuje vremensku evoluciju Schroedingerove jednadžbe upravo samoadjungiran operator. U konačnici postavljamo model atoma vodika i dokazom Kato-Relichovog teorema pokazujemo da se upravo radi o samoadjungiranom operatoru, te dajemo karakterizaciju spektra.
Abstract (english)
In this thesis, we present an outline of the theory of unbounded self-adjoint operators acting on Hilbert space and their application to the hydrogen atom. We show that the main results of operator theory, including Spectral theorem and Stone’s theorem, apply exclusively to self-adjoint operators. In practical application, operators we come across are often symmetric. We show necessary conditions for symmetric operators to be self-adjoint and we explore the possibilities of extending symmetric operators to self-adjoint operators. Our main examples include position and momentum operators. Afterward, we briefly talk about the Spectral theorem as one of the most important results in functional analysis, which for arbitrary self-adjoint operator A allows us to define \(f\) (A) for any Borel function \(f\) . Using Spectral theorem we can prove Stone’s theorem, which shows that generator of a one-parameter unitary group that characterizes the time evolution of Schr¨odinger equation is self-adjoint. Lastly, we describe the simple model of the hydrogen atom, characterize its spectrum, and using Kato-Rellich’s theorem show that the governing operator is self-adjoint.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:100660
Study programmeTitle: Applied Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-11-02 14:41:08