Thanks for the additional answer. So $H$ is actually an operator?

@ThePointer, yes, I thought you knew this (from reading your question). The classical Hamiltonian (total energy) for a particle in potential is $H = \frac{p^2}{2m} + V(x)$. The QM Hamiltonian operator is $H=\frac{P^2}{2m} + V(X)$ where $P$ is the momentum operator and $X$ is the position operator. On the position (wavefunction) basis, the Hamiltonian operator takes the form in your question. TDSE is a non-trivial statement about the Hamiltonian operator: the Hamiltonian is the generator of time evolution

"So the wavefunctions we seek satisfy $\frac{\partial}{\partial t}\psi_E(x,t) = -i\frac{E}{\hbar}\psi_E(x,t)$" Why is this the case? I don't understand your reasoning here.

@ThePointer, that's just the result of combining the 1st and 3rd equations, i.e., $H\psi_E(t)=E\psi_E(t)=i\hbar\frac{\partial}{\partial t}\psi_E(t)$

@ThePointer, no, that's not how it works. Equation 3 is general (holds for any $\psi$). Equation 1 holds only those $\psi$ that are eignenfunctions of the energy operator, i.e., that 'pass through' the operator with scaling factor $E$. The subscript is just a label to help keep everything straight.

@ThePointer, note that it's more or less customary to label state vectors (whether an abstract ket or a position basis wave function) with their eigenvalue if they are eigenkets of an observable. For example, using abstract kets, you'll see something like $N|n\rangle=n|n\rangle$ or $X|x\rangle=x|x\rangle$. A general ket, on the other hand, is often denoted by $|\psi\rangle$

Hmm, ok. I now understand all of the mathematics in your answer, but I think that, since I am a novice to quantum mechanics, I am not understanding the physical point that you're trying to make. I now understand everything you did here, but if I'm not mistaken, there is a deeper physical point about the energy $E$ that you're trying to make here, and that is lost on me (again, likely because I have not yet developed the necessary level of physical understanding of the mathematics).

Ok, thanks for the clarification. At the moment, as you probably saw in my other (now deleted, since it was apparently off-topic) question, I am now trying to understand how to derive the general solution of the TISE. Unfortunately, the documents I come across are lengthy discussions on the Shrödinger equation in general, and do not seem to go into detail on deriving the TISE general solution (example: homepage.univie.ac.at/reinhold.bertlmann/pdfs/…).

Have you had a course in differential equations yet?

Yes, mostly ODEs with some PDEs.

It has been a while since I have used my ODE knowledge, so I need to review things as I go along

The problem with the TISE is that I can't find any documents that actually derive it step-by-step

They all seem to just jump straight to the general solution, without taking the time to derive it.

OK, the TISE with fixed potential is just a linear, second order ODE. So, as usual, you try the solution psi(x) = e^rx and find that there are two values of r that satisfy the ODE (which is what you expect - a 2nd order ODE like this should have two independent solutions). The general solution is just an arbitrary linear combination of the independent solutions. Does this sound familiar to you?

Yes. But the point is that I am looking for the derivation of the general solution. Every document that results from a Google search for the general solution of the TISE is either a derivation fo the TISE itself, rather than a general solution, or it is a derivation of the time-dependent solution, which I have obviously already done.

It might be that the only option is to sit there are try to derive the general solution myself.

Anyway, thanks for the clarification. I will review my linear second-order ODE knowledge.

I'm sorry, I don't understand what you're looking for then. By "general solution of the TISE", what do you mean? Do you mean the general solution to the TISE for a specified potential? Or are you thinking of some kind of meta solution for the TISE with unspecified potential?

For a uniform potential. If I am not mistaken, this gets us $\psi(r) = Ae^{ikz} + Be^{-ikz}$

All I've been looking for is a derivation of this.

Apparently, $k^2 = \dfrac{2m}{\hbar^2}(E - V)$ is then found by substitution back into the time-independent Shrödinger equation.

I'm just trying to derive this stuff for myself for educational purposes, and to convince myself that it is correct.

OK, I'm clear on that now (just wanted to make sure we're on the same page). So, as outlined earlier, this solution is found by inserting the ansatz psi(x)=exp(rx) into the ODE. When you do this, you get a quadratic equation in $r$. This equation is know as the characteristic equation. It has two solutions since it is a quadratic. You've seen this before, correct?

If you haven't (or don't remember), take a look at this and let me know if it helps: math24.net/…

Yes, I remember working with this many times

Thanks for the clarification.

When I first saw it, I remembered that there was some ODE or PDE solution method that produced a superposition of waves, but it had been a while since I'd done any work with differential equations, so I couldn't remember exactly what method it was, nor could I remember how to apply it

All I was looking for is an explanation or document to remind me of the specifics, but this has ended up being much more difficult than anticipated.

Anyway, thanks for the clarification. I will now work through it myself

Great!