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These notes offer a basic introduction to the primary mathematical conceptsof quantum physics, and their physical significance, from the operator andHilbert space point of view, highlighting more what are essentially theabstract algebraic aspects of quantisation in contrast to more standardtreatments of such issues, while also bridging towards the path integralformulation of quantisation. A discussion of the (first) Noether theorem andLie symmetries is also included to complement the presentation. Emphasis is putthroughout, as illustrative examples threading the presentation, on the quantumharmonic oscillator and the dynamics of a charged particle coupled to theelectromagnetic field, with the ambition to bring the reader onto the thresholdof relativistic quantum field theories with their local gauge invariances as anatural framework for describing relativistic quantum particles in interactionand carrying specific conserved charges.

A Pedestrian Introduction to the Mathematical Concepts of Quantum Physics