Quantum Mechanics was first conceived at the turn of the twentieth century, and since has shook the foundations of modern physics. It is a radically different viewpoint from classical physics, which works on the macroscopic scale, in contrast to quantum mechanics' microscopic domain. Though at first it was heavily debated by members of the scientific communit, it is and has been both theoretically and experimentally verified by the likes of Einstein, Heisenberg, Shr\"{o}dinger, to name but a few. This being said, it is still an incomplete theory, and has yet not been concretely proved, despite strong experimental evidence for its truth. The aim of this report is to introduce the field of quantum mechanics, and to investigate the notions of conservation/symmetry, familiar from classical mechanics. The transformations we consider here are parity/space-inversion, lattice translation and time reversal. We will build a knowledge base by analysng the operators that represent these transformation within a quantum mechanical framework. This paper is presented for an audience that has completed a mathematics degree course up to and including second year. The specific feilds we draw upon include differential equations (MA1OD1, MA2OD2, MA2PD1), linear algebra (MA2LIN), and dynamics (MA2DY). These modules are assumed to be prior knowledge. The main sources of information for this project are:
An Introduction To Quantum Mechanics, D.J. Griffiths (1995), Second edition, Pearson Education ltd., 2005
Modern Quantum Mechanics, J.J. Sakurai (1994), First edition, Addison-Wesley Publishing Company inc. 1994
which are referenced throughout. For specific pages, see the bibliography, which is found in section 6.

Chris