Our aim in this paper is to give an overview of recent results concerning randomization methods applied to Schrödinger equations obtained in [33, 32, 35]. We focus here on properties of the harmonic oscillator and Her-mite functions for simplicity. The general idea is that several well known deterministic results like Sobolev type inequalities are improved in a large extent by introducing some random-ness. This is very useful in particular for N.L.S with supercritical initial data.