Path integrals and symbols of operators in Quantum Mechanics

This primer was presented as an advanced undergraduate and postgraduate course at University College Dublin as a follow up to my first series of lectures on Quantum Mechanics within the traditional formalism of operators and wave functions. In turn these notes rely on the lectures given by my MSc and PhD theses supervisor and scientific collaborator Professor N.A. Sveshnikov (1953 - 1997) at the Department of Physics of Moscow M.V. Lomonosov State University attended by myself as a student. Proceeding from the seminal ideas of P.A.M. Dirac and R. Feynman and based on the first principles of Quantum Mechanics we derive the path integral representations for the evolution amplitude, the partition function and other observables averages. We compare different schemes of quantisation yielding corresponding symbols of operators in the phase space.