Clarification on Schrodinger's Equation

[uestions]

Are my thoughts correct? **Wave function just means the wave function psi. I will specify when the wave function is squared.

1.) Schrodinger's Equation describes particles-their position, energy, spin (through the "numbers" l, n, and m).
2.) Simplified, SE says the total energy is the sum of kinetic and potential energies of a system.
3.) Wave functions (psi) are the solutions to SE. Wave functions produce n, l, and m, and psi squared gives the probability of a particle being in a specific position.
4.) Wave functions have no significance until they are squared.
5.) Wave functions don't actually map out, point by point, or crest by crest, the possible path of a particle.
6.) Kinetic energy is directly proportional to psi; potential energy is inversely proportional to psi.

 

[Nugatory]

#5 and the first sentence in #3 are correct.
#4 is correct or not according to exactly what you mean by "significance". The unsquared amplitude cannot be directly measured (does that mean it has no significance?), but it is an interesting and important quantity and we work hard at calculating it because we need it (does that mean that it is very significant?).
The rest are not (and #4 and #6 contradict one another).

The l/n/m quantum numbers only appear when we apply Schrodinger's equation to one particular problem, a particle moving in a central-force potential. That's a very important problem (much of modern chemistry is built on it) but there are many more problems in which we use Schrodinger's equation to describe the evolution of the wave function without the l/n/m numbers ever making an appearance.

It's misleading to think of the quantum-mechanical Hamiltonian operator as just the sum of kinetic and potential energy, or the E operator ##i\hbar\frac{\partial}{\partial{t}}## as the total energy; neither ##H\psi## nor ##E\psi## are numbers so they can't be energies (although they are related to the energy and you can use them to calculate the expectation value of the energy in a given system). It's best to think of them as abstract mathematical objects that we use to construct the differential equation that we then solve to find ##\psi##.

Your best bet here may be to work through a first-year QM textbook. That can be a very frustrating process if you're doing it on your own, but there are plenty of people here who can help you through the hard spots.