The final chapter of this report studies the so called instanton solution as a way of describing quantum mechanical vacuum tunneling effects as a semi-classical problem. Following on from this, we calculate a general formula for the quantum mechanical propagator in terms of path integrals over different homotopy classes. To this end, in chapter 3, we introduce the topological concept of homotopy classes, and apply this to derive the spin statistics theorem, and discuss the phenomenon of parastatistics for systems constrained to dimension d<3. After an overview of the techniques of integration and the relationship to the familiar results of quantum mechanics such as the Schroedinger equation, we study some of the applications to mechanical systems with non-trivial degrees of freedom and discuss the advantages that path integration has as a formulation in studying these systems. In this report, we deliver a detailed introduction to the methods of path integration in the focus of quantum mechanics.
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