  These notes offer a basic introduction to the primary mathematical concepts
of quantum physics, and their physical significance, from the operator and
Hilbert space point of view, highlighting more what are essentially the
abstract algebraic aspects of quantisation in contrast to more standard
treatments of such issues, while also bridging towards the path integral
formulation of quantisation. A discussion of the (first) Noether theorem and
Lie symmetries is also included to complement the presentation. Emphasis is put
throughout, as illustrative examples threading the presentation, on the quantum
harmonic oscillator and the dynamics of a charged particle coupled to the
electromagnetic field, with the ambition to bring the reader onto the threshold
of relativistic quantum field theories with their local gauge invariances as a
natural framework for describing relativistic quantum particles in interaction
and carrying specific conserved charges.
