I have always been confused by the relationship between the Schrödinger equation and the wave equation.
iℏ∂ψ∂t=−ℏ22m∇2+Uψ-vs-∇2E=1c2∂2E∂2t
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 δ∫nds=0 and Maupertuis least action principle δ∫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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