delo diplomskega seminarja
Žan Hafner Petrovski (Author), Pavle Saksida (Mentor)

Abstract

V tem delu se spoznamo s Sturm-Liouvilleovo teorijo, ki predstavlja močno orodje za obravnavo različnih problemov. Diferencialne enačbe, ki pogosto nastopajo v modelu nekega fizikalnega pojava, lahko prevedemo na problem iskanja lastnih vrednosti in lastnih funkcij diferencialnega operatorja. Izkaže se, da je sistem lastnih funkcij pri določenih pogojih kompleten, zato lahko rešitve začetne diferencialne enačbe izrazimo kot linearne kombinacije lastnih funkcij. Tak način reševanja diferencialnih enačb najprej demonstriramo na primeru Besselove enačbe, ki jo dobimo iz večrazsežne valovne enačbe, potem pa se dotaknemo še osnovnih pojmov kvantne mehanike in prek metode separacije spremenljivk rešimo Schrödingerjevo enačbo za problem kvantnega harmoničnega oscilatorja.

Keywords

matematika;Sturm-Liouville;kompletnost;Besselova enačba;kvantna mehanika;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [Ž. Hafner Petrovski]
UDC: 517.9
COBISS: 18821465 Link will open in a new window
Views: 1162
Downloads: 168
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Other data

Secondary language: English
Secondary title: On Sturm-Liouville theory
Secondary abstract: In this paper we discuss the Sturm-Liouville theory, which has proven to be a useful tool when dealing with a variety of problems. Differential equations that often present themselves when modelling physical phenomena can be reduced to the problem of finding eigenvalues and eigenfunctions of a differential operator. It happens to be that the system comprised of all eigenfunctions is complete under certain conditions and that, therefore, each possible solution of the differential equation can be expressed as a linear combination of the eigenfunctions. We demonstrate this method of solving differential equations in the case of the Bessel equation, which we derive from the multidimensional wave equation. We also acquaint ourselves with the very basics of quantum mechanics and via the method of separation of variables solve the Schrödinger equation for the problem of quantum harmonic oscillator.
Secondary keywords: mathematics;Sturm-Liouville;completeness;Bessel equations;quantum mechanics;
Type (COBISS): Final seminar paper
Study programme: 0
Embargo end date (OpenAIRE): 1970-01-01
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages: 33 str.
ID: 11228327