In this thesis, we focus on the search of solutions to the Schrödinger Equation that are, for any fixed time, the so-called "chirp" functions. Namely, we look for explict solutions, which take the form of superposition of "chirps", to both the homogeneous and the non-homogeneus Cauchy Problem for the Free Particle Schrödinger Equation and the Quantum Harmonic Oscillator Schrödinger Equation, as well. We point out that the proofs of our results, just based on classical differential calculus, perfecty match with the latest ones on the propagation of the Gabor wave front set for the Schrödinger Equation, founded on Pseudo-differential Theory.