Fourier Transforms and the Solution of the Schrödinger Equation

We had arrived at the Schrödinger equation starting from the wave packet

$$\Psi(x,t) = \int_{-\infty}^{\infty}dk a(k) e^{i(kx-\omega(k) t)},\quad \hbar\omega(k)=E=\hbar^2k^2/(2m).$$ (42)

In the following, we formalize such superpositions of plane plane waves by introducing the concept of Fourier integrals and the Fourier transform of a function. The Fourier transform is a powerful tool to solve linear partial differential equations such as the Schrödinger equation for a free particle (potential $V(x)=0$).