Jednočestični Schrödingerovi operatori

master's thesis

• Title (english): One-particle Schrödinger operators
• Author: Valentino Novak
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Zoran Vondraček (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-24, 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: samoadjungirani operatori; Hilbertov prostor; Spektralni teorem; Stoneov teorem; Borelova funkcija; Kato-Relichov teorem
• Keywords (english): self-adjoint operators; Hilbert space; Spectral theorem; Stone’s theorem; Borel function; Kato-Rellich’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:100660
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-11-02 14:41:08