This question comes from and is on the border of physics, but I ask it here to seek possible connection to a more general mathematical framework and/of references to existing literature.
For a given complex valued $\psi(\omega)$, let us define two quantities:
Fourier transform: $\tilde{\psi}(t)\equiv \int \psi(\omega) \, e^{- i \omega t} d \omega$
Second moment of the complementary variable: $\Delta t^2 \equiv \int_{-\infty}^{+\infty} | \tilde{\psi}(t)|^2 t^2 \, dt - \left( \int_{-\infty}^{+\infty} | \tilde{\psi}(t)|^2 t \, dt \right)^2$
I believe I can prove the following theorem:
For a given non-negative real-valued function $p(\omega)$, among all complex-valued functions $\psi(\omega)$ such that $|\psi(\omega)| =p(\omega)$, the minimal value of $\Delta t$ is attained for $\psi(\omega)=p(\omega) e^{- i \omega t_0}$ with an arbitrary constant $t_0$.
Question: what is the correct name and a reference for the above theorem?