I'm new to quantum mechanics and I am currently trying to understand finite potential well (although my question is not specific to finite potential well ). In the Schrodinger equation, many texts and internet sources say that $V(x)$ is the potential. I have the following queries regarding this claim.
a) When they say $V(x)$ is the potential, is it just a way of "meaning" potential energy instead? Because after all, schrodinger equation is an energy conservation equation with kinetic and potential energies.
b) Secondly, if $V(x)$ means potential energy, then shouldn't the potential energy be dependent on the particle under consideration (I know a similar question has be asked before on this platform, but I am not convinced by the answer).
c) And if $V(x)$ is the potential energy, is it the potential energy that the particle under discussion would have had it been classically placed at the point $x$ or does $V(x)$ have the same meaning as it is traditionally defined, where a positive unit charge is placed at the point and it is the energy of that positively charged particle?
d) Lastly, when we are solving the finite potential well, we find the wave function for the two cases where $E<V_0$ and $E>V_0$. Are these two cases for the same particle under discussion or are we saying "**if there is a particle whose energy $E$ is greater than $V_0$ (or if there is a particle whose energy $E$ is less than $V_0$) **