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.

Improving business processes in a globally dispersed make-to-order supply chain. Case study on outbound logistics in a Multinational Corporation

The purpose of this thesis is to find out how outbound logistics process can be improved by reducing unnecessary waste in a globally dispersed make-to-order (MTO) supply chain. The research problem was addressed by a multinational corporation that aims to find a solution for reducing unnecessary waste in their outbound logistics process. The focus is on customized products that are delivered via sea transportation. Theoretical framework for improving outbound logistics processes in globally dispersed MTO supply chain was created based on business process management, Porter’s value chain theory, value stream mapping and current reality tree. The empirical research was conducted by using constructive approach due to its ability to research a practical problem and to improve the existing practices. The data was collected from ten semi-structured interviews and three non-participant observations. By analysing the data and applying the theoretical framework, five types of waste were detected in the process that were seen to derive from six root causes. Practical solution was constructed to reduce the waste in the process by combining the existing literature with the ideas raising from empirical data. The results of this thesis suggest that a MNC with a globally dispersed MTO supply chain can improve its outbound logistics process by applying activities that enhance internal and external integration, collaboration and coordination, and increase predictability of the process. This research has practical relevance both for the case company as well as for other MNCs with globally dispersed MTO supply chains that aim to improve their outbound logistics processes. This research contributes to the BPM and CRA research by providing an evidence for their applicability in the new context.