Vector spaces, Hilbert space. Linear Functionals, Dirac's Functional and the Green Functions. Linear operators, bounded operators, hermitian and unitary operators, the spectrum of operators. Classical Physics, Waves mechanics, The wave differential equation. Experiments showing the inadequacy of Classical Mechanics. Quantization of energy states, the wave - particle dualism of matter, Uncertainty principles. The fundamental postulates of Quantum Mechanics, description of states and observables, the quantum law of motion. Continuity equation. The Heisenberg representation, the matrix mechanics. Quantum mechanical problems, stationary states, wavepackets, free particle, piecewise constant potentials, the linear harmonic oscillator.