# Why does the wave function of a non relativistic particle flatten out over time?

The Hamiltonian I used is the classical one with no potential energy: H=p^2/2m $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}$$ I want to gain an intuitive understanding of what's happening in this differential equation.

• The discussion at the end of the diffusing Gaussian wavepacket is not clear to you? It details how it is driven by the uncertainty principle and basic properties of Fourier Analysis. – Cosmas Zachos Jul 2 '19 at 16:07
• Please don't call the quantum Hamiltonian "classical"! – electronpusher Jul 2 '19 at 16:14
• – Cosmas Zachos Jul 2 '19 at 18:37