The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Conflicting Memory in Estonia: How Do Estonian Russian-speakers Understand the Events of 1939-40?

This thesis studies the collective memory of the Russian-speaking minority living in Estonia. The minority is exposed to two national narratives regarding the incorporation of Estonia into the Soviet Union in 1940. According to the Estonian narrative, Estonia was occupied and annexed by the Soviet Union while the Soviet-Russian narrative sees the actions to have been legal and voluntary. This thesis firstly examines thoughts the existence of these two opposing narratives evoke among the Russian-speaking minority and secondly it explores whether the views of the minority compare with the two official yet divergent narratives. The study focuses on the second and third generation minority members. The topic belongs to the field of memory studies. The objective is to understand the views the Russian minority have towards the controversial events of the years 1939-40. To accomplish the objectives set, a web-based survey using open-ended and multiple-choice questions was conducted. The open-ended questions addressed the main research questions while the multiple-choice questions contributed to forming a more comprehensive understanding of the subject in question. In order to interpret the data, qualitative content analysis has been applied. Based on the findings, the Russian-speaking minority respondents’ understanding of the events of 1939-40 could be described as fragmented, inconsistent and including viewpoints that resulted from the merger of different storylines. There is no single cohesive or coherent narrative of the past amongst the minority. In addition to that, their views do not generally comply with the narrative of the Russian Federation as often referred to in literature, even though the minority respondents do not want to see the Soviet involvement as critically as the Estonian narrative does. Many respondents conceive the events of 1939-40 as ambiguous revealing the ability to be tolerant and receptive in their views regarding the past.