The wave function and the Shrodinger equation

[DiracPool]

I've been a bit puzzled regarding the relation of the Schrödinger equation (SE) and the wave function, and what they mean. Here's kind of where I'm at.

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 only a single one 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.

 

[atyy]

The wave function is the state of the system. It can be essentially any function.

Given an initial state, the Schroedinger equation gives the time evolution of the state until a measurement is made.

When you make a measurement, you get an outcome and the wave function collapses. After the collapse, the system again evolves according to the Schroedinger equation.

Here are some more details about observables and operators in measurements. When you measure an observable, say position, there will be a corresponding operator. In this case, since we picked position, the corresponding operator will be the position operator. The measured value of position will be an eigenvalue of the position operator, and the system will collapse into the corresponding eigenfunction of the position operator. (That is strictly wrong, but the rough idea is right. It is strictly wrong, because the position eigenfunction are not a strictly allowable states of the system. It is roughly right since if one picks continuous variables there are mathematical subtleties that are conceptually incidental from the physical point of view.)

How does one find out the wave function that an unknown but reliably repeated procedure prepares? That in general is hard, and is the subject of quantum state tomography.

Eigenfunction, eigenvector and eigenstate all mean the same thing. There are some differences, but anyone who understands the subtleties will know the differences are semantic.

 

[atyy]

Incidentally, it is the time-dependent Schroedinger equation (TDSE) that is important because it determines the time evolution of an abitrary state of the system.

The time-independent Schroedinger equation (TISE) is overall less important. The main importance of the TISE is that it is the equation that determines the eigenvalues and eigenvectors of the energy operator - also known as the Hamiltonian.

The second reason the TISE is important is that it is obtained from the TDSE by separation of variables. So given an arbitrary initial wave function, you can write that as a superposition of energy eigenfunctions. The time evolution that the TDSE gives of the initial wave function is the same as if each energy eigenfunction was multiplied by something like exp(iEt) (I always get signs and Planck's constant wrong here).

A good introduction to QM is http://arxiv.org/abs/1007.4184 (except for the last chapter, in which case look at Landau and Lifshitz, and Weinberg).

 

[DiracPool]

A good introduction to QM is http://arxiv.org/abs/1007.4184 (except for the last chapter, in which case look at Landau and Lifshitz, and Weinberg).

Thanks for the reference atyy. I've already started to read it. Looks good. Although..have you checked out page 16 where he says:

Quantum mechanics learns us that this is not correct.

Lol. What's that all about? I remember that as the iconic SAT question.. Which sentence is correct:

A. "He taught me the correct procedure"
B. "He learned me the correct procedure"

 

[atyy]

OK, maybe these notes by John McGreevy use English that is more standard: http://physics.ucsd.edu/~mcgreevy/w14/lecture-notes.html :)

Actually, McGreevy's notes look superb. But the order of material is quite different from Gaasbeek's. So probably one has to read up to the end up Gaasbeek's chapter 4 before looking at McGreevy's notes, if one is starting from the wave function.

 

[DiracPool]

OK, maybe these notes by John McGreevy use English that is more standard:

I'm not really worried about the English. I just thought that one line was funny, I've never seen that phrase in print like that. As long as it's fairly legible and the physics explanation is good, that's all that matters to me, and from what I've read so far, the Gaasbeek paper looks great. Thanks for the McGreevy paper too, the more the merrier. I'll have to get to that tomorrow.

As far as my point about the stand-alone ψ term in the SE being a "dummy" variable or function with no value, did I get that right? I think so, because now that I come to think of it, that's really what a differential equation (DE) is, isn't it? We set up a differential equation because we know some properties of the system but we don't know the function we are looking for. So we we set up a DE and just kind of stick the symbol for the function we are looking for in there, even though it has no value at the time, and hope that we can solve the DE and find out what the function is. (you can tell I'm not a mathemetician )

This is essentially what's going on here with the Shroedinger equation, right? ψ is the unknown function we're looking for and the SE is the differential equation we've constructed to find it?

 

[Nugatory]

This is essentially what's going on here with the Shroedinger equation, right? ψ is the unknown function we're looking for and the SE is the differential equation we've constructed to find it?

yes.

 

[vanhees71]

A good introduction to QM is http://arxiv.org/abs/1007.4184 (except for the last chapter, in which case look at Landau and Lifshitz, and Weinberg).

Skip the introduction too. He's one of those who copy the very misleading description of the photoeffect on p. 13ff!

 

[atyy]

This is essentially what's going on here with the Shroedinger equation, right? ψ is the unknown function we're looking for and the SE is the differential equation we've constructed to find it?

Yes, for the case of the time-independent Schroedinger equation (TISE), Hψ=Eψ.

H is the energy operator, also known as the Hamiltonian. The TISE is the eigenvalue equation for the energy operator. In general the energy operator will do "rotate" the wave function. However, the TISE asks for the particular wave functions that the energy operator does not rotate, but only multiplies by a constant, called the eigenvalue. The particular wave functions that are solutions of the TISE are the eigenfunctions of the energy operator, and each eigenfunction has its own eigenvalue.

The main physical significance of the eigenvalues and eigenfunctions of the Hamiltonian are that when one measures the energy, the possible measurement result is one of the eigenvalues of the Hamiltonian. After the measurement, the wave function will change from whatever it was before the measurement into the eigenfunction corresponding to the eigenvalue that was the measurement result.

Of less physical significance, and more of technical significance, is that solving the TISE is an intermediate step in solving the TDSE.

 

[DiracPool]

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).

Btw, why is the second term in that solution, A2 exp^(ikx)? Isn't just ψ(x)=A1 exp^(-ikx) alone a solution to the free particle equation? i. e. ψ''(x)=-k^2ψ

Did they just add the A2 expression because it is also a solution and allows for more terms to use in using the boundary conditions in coming up with the final solution, ψ(x)=√(2/a) sin(nπx/a), or is ψ(x)=A1 exp^(-ikx) + A2 exp^(ikx) the only possible solution to the SE for the free particle. (again, forgive me, I'm not a mathematician).

 

[vanhees71]

First, you look for the most general solution. A 2nd order linear homogeneous ODE has two linearly independent solutions, whose linear combinations span the full space of solutions. Then you need initial our boundary values to fix the corresponding two coefficients to the special solution you need to solve your physics problem.

 

[DiracPool]

First, you look for the most general solution. A 2nd order linear homogeneous ODE has two linearly independent solutions, whose linear combinations span the full space of solutions. Then you need initial our boundary values to fix the corresponding two coefficients to the special solution you need to solve your physics problem.

Ahh, ok, that's right, now I remember. Been a while since I took diff. eq. Thanks.