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O Sturm-Liouvilleovi teoriji : delo diplomskega seminarja
ID Hafner Petrovski, Žan (Author), ID Saksida, Pavle (Mentor) More about this mentor... This link opens in a new window

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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.

Language:Slovenian
Keywords:Sturm-Liouville, kompletnost, Bessel, kvantna mehanika
Work type:Final seminar paper
Typology:2.11 - Undergraduate Thesis
Organization:FMF - Faculty of Mathematics and Physics
Year:2019
PID:20.500.12556/RUL-110731 This link opens in a new window
UDC:517.9
COBISS.SI-ID:18821465 This link opens in a new window
Publication date in RUL:19.09.2019
Views:1516
Downloads:201
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Secondary language

Language:English
Title:On Sturm-Liouville theory
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.

Keywords:Sturm-Liouville, completeness, Bessel, quantum mechanics

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