Energetické spektrum částice v trojrozměrné pravoúhlé potenciálové jámě nekonečné hloubky z hlediska teorie čísel

Abstract

In this bachelor thesis we will be concerned with mental model of a particle enclosed within impenetrable container, which we call potential well. The text begins with brief introduction to the topic, followed by some basic concepts and the introduction to quantum mechanics, during which we will get acquainted with Schrödinger equation, which will be solved then for above mentioned model. This way, we will recieve discrete spectrum of energies of a particle, which is, by suitable choice of units, equivalent to set of numbers, expressible as a sum of three squares of natural numbers. We will take a close look at this set and get acquainted with issues of sums of integer squares and asymptotic density. In the end of the thesis, we will construct mental experiment with energy measuring and, on the basis of the acquired results, we will try to find out the count of particles in the well using statistical methods.