I was wondering, if anyone knows the origins of how Schrödinger arrived to his equation? And can it be derived from Newtonian mechanics? How did Schrodinger form the equation out of his MIND?
I also I was asking because I was playing around with Newtonian mechanics, I believe I derived the Schrödinger equation...
It can be derived from the classical wave equation for photons and the de Broglie wavelength. Suppose you have an electromagnetic wave $$\psi = Ae^{i (\mathbf{k} \cdot \mathbf{r} - \omega t)}.$$ Taking the spatial derivative yields $$\nabla^2 \psi = - k^2 \psi.$$ Since $\hbar k = p$, $$-\frac{\hbar^2}{2m} \nabla^2 \psi = \frac{p^2}{2m}\psi.$$ Taking the time derivative yields $$\frac{\partial \psi}{\partial t} = -i \omega \psi.$$ Since $\hbar \omega = E$, $$ i \hbar \frac{\partial \psi}{\partial t} = E \psi.$$ The energy is $$E = \frac{p^2}{2m} + V.$$ Hence $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V\psi.$$
