I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of c, the speed of light. A simplistic argument can be- $$v_{phase}=\frac{\omega}{k}=\frac{E}{p}=\frac{mc^2}{mv_{particle}}=\frac{c^2}{v_{particle}}$$ (using de Broglie's formula and Planck's formula) Since $v_{particle}<c$, hence $v_{phase}>c$. Thus we use not a simple wave, but instead a superposition of multiple waves, which represents a wave packet. Wave packets, as far as I know, should have a varying amplitude.

While deriving Schrodinger equation, why don't we treat the amplitude $A$ as variable quantity? Am I missing something?

I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of $c$, the speed of light. A simplistic argument can be- $$v_{phase}=\frac{\omega}{k}=\frac{E}{p}=\frac{mc^2}{mv_{particle}}=\frac{c^2}{v_{particle}}$$ (using de Broglie's formula and Planck's formula) Since $v_{particle}<c$, hence $v_{phase}>c$. Thus we use not a simple wave, but instead a superposition of multiple waves, which represents a wave packet. Wave packets, as far as I know, should have a varying amplitude.

While deriving Schrodinger equation, why don't we treat the amplitude $A$ as variable quantity? Am I missing something?