I have always been confused by the relationship between the Schrödinger equation and the wave equation.
$$ i\hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m} \nabla^2+ U \psi \hspace{0.25in}\text{-vs-}\hspace{0.25in}\nabla^2 E = \frac{1}{c^2}\frac{\partial^2 E}{\partial^2 t} $$
Because of the first derivative, the Schrödinger equation looks more like the heat equation.
Some derivations of the Schrodinger equation start from wave-particle duality for light and argue that matter should also exhibit this phenomenon.
In some notes by Fermi, it was derived by comparing the Fermat least time principle $\delta \int n \;ds = 0 $ and Maupertuis least action principle $\delta \int 2T(t) \; dt = 0 $.
Was this ever clarified? How can we see the idea of a matter-wave more quantitatively?
To summarize, I am trying to understand why the Electromagnetic wave equation is hyperbolic while the Schrodinger equation is parabolic.
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