Scattering for the finite square well

[hmparticle9]

Homework Statement

Understanding scattering for the finite square well

Relevant Equations

$$\psi(x) = Ae^{ikx} + Be^{-ikx}$$

$$\psi(x) = Fe^{ikx} + Ge^{-ikx}$$

Consider a square well of length $2a$ centred at the origin. With a constant potential in the well $-V_0$. That is, the potential function is an even function. Consider the solution to the problem outside the finite square well. First the left hand side:
$$\psi(x) = Ae^{ikx} + Be^{-ikx}$$
The right hand side:
$$\psi(x) = Fe^{ikx} + Ge^{-ikx}$$
Since the potential is symmetric, we can assume that the $\psi$ is either even or odd. Let's assume that it is even. Then $A = G$ and $B=F$.

To me, looking at a diagram, it appears that we have waves of equal amplitude $A$ == $G$ "coming in" to the well and waves of equal amplitude leaving the well ($B==F$). Surely this must mean that $A == G == B == F$? For otherwise one of the outgoing waves would have a larger amplitude than the other...

looking at the solutions, I am incorrect. I understand the solutions, but I do not understand where my logic is failing and why.

Another point is that the solutions do not consider the odd case for $\psi$. Would this be because the waves cancel each other out? But within the barrier the odd solution would "correct" the waves? I am not sure.

 

[hmparticle9]

On thinking about it, my conclusion A == B == C == D implicitly makes the assumption about how much of the incoming waves are reflected/transmitted. Basically I have assumed that all that goes in must all come out at the same "time". Am I on the right track here? For instance a proportion of the waves might get "stuck" in the well. But I am still not clear as to why the "odd" case was not considered in the solution.

 

[Steve4Physics]

I'm (very much!) no expert so this is a pretty superficial explanation but might help.

region 1 $V=0$ | region 2 $V = -V_0$ | region 3 $V=0$

You are (presumably) considering the specific case of an incident particle in region 1 moving right. The possible outcomes are transmission (into region 3) and reflection (back into region 1).

There is no possibility of the particle existing in region 3 and moving left, so in your equations you can make the simplification that $G= 0$.

The solution is no longer symmetrical about x=0. As a consequence, I believe that $\psi$ inside the well can be odd, or even or asymmetric - you don’t need to know. All that’s required is continuity (of $\psi$ and $\frac {d\psi}{dx}$) at the boundaries.

There's quite a decent video here.

 

[hutchphd]

On thinking about it, my conclusion A == B == C == D implicitly makes the assumption about how much of the incoming waves are reflected/transmitted.

It is worse than that. If you assume the problem is symmetric, then the boundary conditions at infinity (+/- x) are also assumed symmetric and so there is no net flux. So carefully define what you wish to know and what the boundary conditions are. To obtain an answer you will need to actually solve the problem for all three regions (including in the well) by matching at the boundaries. This will allow you to look at the boundary values appropriate to the scattering event that you want. This all becomes more difficult because the continuum states are not mormalized in the usual way. I think most texts do this part of scattering very badly.......it is confusing.. Solving for an asymmetric step potential of size V₀ might be a useful preparatory exercise.
This is an important and non trivial problem which recurs often. You need to spend some detailed time on it.
Also fun! Resonances in the continuum......Incidentally I just looked at the video recommended by @Steve4Physics and do not much like it. Why does he mix complex exponentials with sin and cos at the outset????? Just use complex exponentials (bite the bullet!) His discussion is on the money.

 

[hmparticle9]

Thank you! The only thing that was holding me up was that (LEAP) in logic.

 

[hmparticle9]

One more thing. In the solution to the problem they say that $A = G$ and $B = F$. Does this follow from the fact that the potential is "even"?

 

[hutchphd]

I am not sure of the solution to which you allude. There are a host of different ways to solve this, depending upon how you treat the asymetry imposed by "a particle starts from the left at early times" which can get a bit tricky. So particular questions will evoke particular answers. I do think you should work it through in detail. It will likely take a few hours hard work. And prudent use of relection symmetries will make it much easier

 

[hmparticle9]

I have gone through the problem in painful detail and a few just like it. I am all for going through things in painful detail, but when the author hints at a more elegant solution, I am all for understanding that too. I just want to understand each step of the more elegant solution provided by the author. I will give you the exact problem and the solution.

Problem: "Construct the S-matrix for the finite square well. This requires no new work, if you carefully exploit the symmetry of the problem."

Here is the S-matrix
https://en.wikipedia.org/wiki/S-matrix

My question is:

Why is the author only considering solutions of the form A <--> G, B <--> F, but not A <--> -G, B <--> -F. There is a theorem in the book that says that even potentials admit even/odd solutions. The statement "Scattering from the left is the same as scattering from the right,..." to me is an acknowledgement that even potential admits even solutions $\psi$. Again, given above.

The constants A,G,B,F being referred to are those in my original post.

 

[hutchphd]

What are eqns. 2.170 and 2.171 ?
Also these assume that the potential is real. The examination of every symmetry is to be encouraged
My point to you was that the solution does not follow from symmetry alone: at some point one needs a solution. This can be S-matrix or wavefunction. It is a matter of taste, but grunt work needs to be done.
(There are some few fine points of asymptotic completeness in S-matrix that always worry me and which I am really not competent to discuss.)
This is a very rich problem and any tiime required is time well spent. The actual full solution is complicated but replete with fundamental physics.

 

[hmparticle9]

Grunt work has been done. Like I said. Just want an understanding of the more elegant solution. Knowing equation 2.170 etc does not help you answer my question about the first statement of the solution. What comes afterwards I understand perfectly well.

"It is a matter of taste, but grunt work needs to be done."

I had completed this problem the long way around which involved a lot of calculation and took a long time, re-deriving many of the results in the book but for the case that we have two incoming waves. Like I said. Just want to fully understand the more elegant solution.