I've been a bit puzzled regarding the relation of the Schrodinger equation (SE) and the wave function, and what they mean. Here's kind of where I'm at.(adsbygoogle = window.adsbygoogle || []).push({});

1) I always thought that in the standard form of the SE, either the time dependent or independent form, that the wave function Psi (ψ), actually had a value in the stand-alone equation. I didn't know what it was, but I suspected it might be something like ψ(x,t)=exp^i(kx-wt).

Now I'm thinking that, in the stand-alone form ψ doesn't have a value at all. It's a dummy variable that has no value until the SE is solved. And that's the whole point of the SE, to find the wave function of the system in question. Is this correct or not?

2) So how do we find the wave function, then, using the SE? If we take a very simple system, say the famous "particle in an infinite square well," we can solve the SE for a well of length 0-a without worrying about a potential energy term. The solution of the TISE ODE in one space dimension is then ψ(x)=A1 exp^(-ikx) + A2 exp^(ikx), where k=√(2mE/hbar^2).

From here we then plug in the constraints or the boundary conditions of the system to eventually yield the wave function ψn=√(2/a) sin(nπx/a), where ψ is say the "system" wave function, and all the individual wave functions you get from plugging in different integer values of n constitute the "eigenfunctions" of the system wave function. Is that right? (Btw, do the terms "eigenfunction," "eigenvector," and "eigenstate" all mean the same thing? Or are there distinctions?)

Also, coupled along with this set of eigenfunctions we derived from the SE is a formula for the energy states, similarly derived from the SE. i.e. En=(n^2 π^2 hbar^2)/(2ma^2), and also quantized via the integer n.

So I guess my question in the big picture is: Is that what the SE is doing? Is it just a mechanism or tool to find out what the wave function for any given system is? My guess is that the main variable that is going to distinguish one system from another and, thus, one wave function from another is the particular potential energy term in the system you are trying to describe. That and the boundary conditions of that particular system. Is that correct?

So then, once we've derived that wave function for a given system, then that wave function tells us or embodies everything that can be known about that system? The way we get this information is to use "operators" on the wave function. What these operators do is act on every one of the eigenstates of the system to give a probability amplitude for the manifestation for each one as a real collapsed state of the entire system in general? And there is onlya singleone of those many eigenstates that the system can collapse into? (in standard Copenhagen interpretation, of course).

The way this all works has puzzled me for some time so any feedback would be kindly welcomed.

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# The wave function and the Shrodinger equation

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