General solution to the Time-independent Schrödinger equation?

[greswd]

Has anyone formulated a general solution to the time-independent Schrödinger equation in terms of the potential function V(r), and if so, what is it?

For any type of V(r).

So, instead of a differential equation, a direct relationship between the wavefunction and the potential.

 

[BvU]

 

[greswd]

View attachment 240135

@BvU
Yes, I tried that first, but I couldn't find anything for, as mentioned, a general solution in terms of V(r).

And by general solution for V(r) I mean any type of V(r).

So I'm hoping some well-read physics experts here might know of it.

 

[BvU]

From the variety of approaches for different kinds of potential functions I estimate a zero probability for a (usable) general solution.
Would be something like a free lunch

some well-read physics experts

The more I think about it, the less I feel qualified

 

[greswd]

From the variety of approaches for different kinds of potential functions I estimate a zero probability for a (usable) general solution.
Would be something like a free lunch The more I think about it, the less I feel qualified

A non-closed form solution would be really good too, hopefully there's one

 

[BvU]

From what I've seen it's the other way around: suppose we have a $\psi$ that satisfies the TISE, what are its specific properties

 

[greswd]

From what I've seen it's the other way around: suppose we have a $\psi$ that satisfies the TISE, what are its specific properties

I believe the TISE generally produces families of discrete solutions, so a general solution would be in terms of both V(r) and parameters like quantum numbers, with the number of quantum numbers depending on the shape of V(r).

 

[BvU]

we say the same thing

 

[greswd]

we say the same thing

nice