In this chapter we describe the Hilbert space formulation of Quantum Physics. The first 10 sections contain general formalisms which apply to all Quantum Physics, the other 6 sections are specific for mechanical quantum systems. The start with a survey of constructions for the Hilbert space of states, then discuss the linear operators acting on this space, their spectral theory and functional calculus. We deduce the Schrödinger equation and the quantum pictures (Schródinger, Heisenberg, and Dirac). We introduce the density matrix and the quantum Liouville equation. Then we proceed to the quantization conditions and canonical quantization. We construct the Schrödinger and momentum representations in \(\mathrm {r}^n\) , and prove the Heisenberg indetermination principle. We also study the Schrödinger representation on a general Riemannian manifold, possibly with boundary. In this context we prove the general quantum Noether theorem and the quantum action principle.