Hi,
I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically...
As an update ; now I'm having problems with boundary conditions. I can't actually enforce the damn boundary conditions, which is problematic. I'll post an update soon. I tried numerically using maple and no matter what I try it always ends up with f being 0 at theta = pi/2.Weeeeeeellll then. Why...
Oh, that should help. I'll work on it and see what I find. I'll post an update soon. Thanks Chestermill (seriously you're fantastic; you've helped me on two problems so far and I really appreciate it).
UPDATE :
So, if we take your ansatz we find that the angular momentum equation is...
So, here's the problem I've come up with that I wanted to solve.
We're going to be using a polar coordinate system for this one. A will represent our angle theta, and r will represent our radial coordinate. We are going to be looking at a non-compressible Newtonian viscous fluids. I'll now put...
Hey. I haven't yet, I'm still at school and my brains fried. I'll post an answer tomorrow morning when my brain is no longer dead.
As a side note; I did find a problem which is amenable to an analytical solution without integrating the radial variation out. It's the problem of a 2d flow using...
Oh, and as an aside while we all try to work things out ; it turns out that the ODE F" + R*F^2 = -C for F(-1) = 0 and F(1) = 0 has existence problems. Numerically solving it is turning out to be problematic.
Thanks so much! I followed most of that, the only thing that I'm confused over is the constant C. So you're saying that you solved Eqn. 3 analytically and that that's how you found the value of C? Thanks again.
That certainly makes sense for the low Reynolds limit. I'm also interested though in deriving the differential equation they derived when inertial contributions are not neglected. I just drove to school and I have a few hours to kill, I'm going to see if I can work on it and get anywhere. I'll...
Well, given that the inertial term goes to zero we arrive at the following : r*dP/dr = u*d^2F(z)/dz^2. In order for this to hold both of these must be equal to some constant, let's call it C. So we get C = r*P'(r), and we get that C = u*F"(z). These are easy ODE's to solve. We find that F(z) =...
* I may be incorrect in some of the things I'm saying here, but I'll try to offer my own explanation.
This is from a purely classical (non quantum mechanical) point of view. In reality we will never see a perfect electromagnetic plane wave. However, because the differential equations governing...
Thanks for confirming that I'm not insane or something. I initially saw this problem and became interested in it. I quickly realized that an analytical solution to the ODE (F" + R*F^2 = -1, F(-1) = 0, F(1) = 0) was not possible or outside the range of my abilities, so I decided to try to find an...
Thanks for your response. That makes sense. You also say that several other assumptions are used to mathematically enforce effects that would not otherwise arise without viscosity. Is the kutta condition such an assumption?
To start this question off, I'll just state what I do understand about potential flow theory
1.) I understand that the navier stokes equations are really difficult to solve analytically except in degenerate cases where the assumptions about the flow cancel out the non-linear terms.
2.) I...
Hello,
So I was reading wikipedia the other day, as I do from time to time. I came across a rather interesting sample problem posed in the article, but seeing as Wikipedia is horrible in some of their physics articles on explaining what's the hell they're doing, I became lost. Here is the...