What is the wave function of a square wave in an infinite potential well?

[Smeags22]

Alright, so here's my problem. I've got a wavefunction between -L/2 and L/2 (symmetric around 0). It's a square wave and it is in an infinite potential well. That's all I know about it. I need to find the wavefunction of it. I was thinking of doing a Fourier sine/cosine series but I'm stuck.

Here's what I've tried:
[itex] f(x) = \Theta (x+L/2) \sqrt(2/L) cos(n\pi*x/L) [itex]
The thought process here was that I needed an f(x) to put into the Fourier series (I'm just going off the wikipedia definition). I have the step function in there because there is only one wave so before -L/2 the wavefunction is 0. But that is what's tripping me up - I don't know how to take the integral of a step function to get my Fourier coefficients. Maybe I don't need that step function. Maybe I don't even need to do a Fourier series. I guess at this point I'm confused and frustrated and need a push in the right direction.

Any suggestions would be greatly appreciated! Thanks in advance!

 

[Smeags22]

Also, how do I get latex to work?

 

[Dick]

I'm not sure your question is making much sense to me. But to use latex see https://www.physicsforums.com/showthread.php?t=8997 The tag [itex] doesn't work.

 

[Smeags22]

I've got a square wave between -L/2 to L/2 in an infinite potential well of the same width. I need to find an equation for this square wave. I'm not sure how to do that.

 

[Dick]

QUOTE=Smeags22;3655033]I've got a square wave between -L/2 to L/2 in an infinite potential well of the same width. I need to find an equation for this square wave. I'm not sure how to do that.[/QUOTE]

If you are given a wavefunction, then that's the wavefunction. There is nothing to find. If you are given a wavefunction at t=0 and you want to find it's time evolution then that's a different story. Is that what it is? Then, yes, you need to find an expression for the wavefunction as a sum of solutions to the Schrodinger equation in an infinite square well potential. Which is indeed a Fourier series problem.

 

[Smeags22]

I don't know the wavefunction for the wave and that's what I need to find. I'm looking for the wavefunction at t=0. I forgot to mention that. I know how to find the solutions in any other situation but the fact that it's a square wave is messing me up.

 

[Dick]

I don't know the wavefunction for the wave and that's what I need to find. I'm looking for the wavefunction at t=0. I forgot to mention that. I know how to find the solutions in any other situation but the fact that it's a square wave is messing me up.

I still don't get it. What do you think is the difference between a 'wave' and a 'wavefunction'?

 

[Smeags22]

Okay, I have a picture of a square wave in an infinite potential box. I need to figure out what the wavefunction is so I can find the expectation value of H. I know how to find the expectation value but I need a psi to be able to do that. I don't know what psi would be for a square wave. I don't know how to make it out of a Fourier series.

 

[Dick]

Okay, I have a picture of a square wave in an infinite potential box. I need to figure out what the wavefunction is so I can find the expectation value of H. I know how to find the expectation value but I need a psi to be able to do that. I don't know what psi would be for a square wave. I don't know how to make it out of a Fourier series.

No! You don't need a psi. You have it. You need to compute the overlap integral of psi with the energy eigenfunctions of the potential well. They form an orthonormal set, yes? It's Fourier series problem.

 

[Smeags22]

And the Fourier series part is where I'm stuck.

 

[Dick]

And the Fourier series part is where I'm stuck.

Can you describe your square wave? You just integrate that times an energy eigenfunction of the square well. It's just an integration problem. Do it with the ground state first.

 

[Smeags22]

No, I don't know what to do with the square wave. The rest of it I get. But I just don't know how to deal with the square wave.

 

[Dick]

No, I don't know what to do with the square wave. The rest of it I get. But I just don't know how to deal with the square wave.

I'll repeat. Can you describe your square wave? Please?

 

[Smeags22]

It's between -L/2 and L/2. That is literally everything I know about it.

 

[Dick]

It's between -L/2 and L/2. That is literally everything I know about it.

Ok, then if I understand that, you integrate 1 times the eigenfunction of the energy between -L/2 and L/2 to get the overlap, right? You don't even need the Theta.

 

[Smeags22]

Why 1? Are you just saying it's got an amplitude of 1 and you're calling that the function?

 

[Dick]

Why 1? Are you just saying it's got an amplitude of 1 and you're calling that the function?

If your function is the square wave that I think it is and I'm guessing, then it's a constant in the potential well. IS it? Multiply that constant times the energy states and integrate. Do the <∅|ψ> thing.

 

[cbetanco]

Maybe your confused with the difference between the potential being square shape (particle in an infinite square well) and what the actual wave function is. If you are not given the wave function, and asked to solve the equation, then just write a solution of the form $\psi=Acos(kx)$ and solve for A and k using restriction on the energy and normalization. If you already know the wavefunction is square, then just time evolve it with shrodinger's eq.

 

[cbetanco]

seems my function got crossed out? Well, just expand in terms of sines and cosines.

 

[vela]

Alright, so here's my problem. I've got a wavefunction between -L/2 and L/2 (symmetric around 0). It's a square wave and it is in an infinite potential well. That's all I know about it. I need to find the wavefunction of it. I was thinking of doing a Fourier sine/cosine series but I'm stuck.

Here's what I've tried:
$$f(x) = \Theta (x+L/2) \sqrt(2/L) cos(n\pi*x/L)$$
The thought process here was that I needed an f(x) to put into the Fourier series (I'm just going off the wikipedia definition). I have the step function in there because there is only one wave so before -L/2 the wavefunction is 0.

For the infinite square well, the wave function vanishes outside the well. I think we're all assuming here that the well is from x=-L/2 to x=+L/2. Is that in fact correct? If it is, the wave function will always be 0 when x<-L/2 or when x>L/2 — that is, ψ(x)=0 outside the well. You really only need to worry about what's happening on the interval [-L/2, L/2].

As far as I can tell from what you've written, the wave function is constant inside the well, so you have
$$\psi(x) = \begin{cases} 0 & x<-L/2 \\ c & -L/2 \le x \le L/2 \\ 0 & x>L/2\end{cases}$$where c is some constant.

Is this what you mean? If not, could you provide a better description that simply repeating the phrase "square wave"?

Also, what exactly are you being asked to calculate? Maybe you don't even need an expression for ψ(x).