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Finite Square-Well Potential

  • #1
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24. Apply the boundary conditions to the finite square-well potential at x=0 to find the relationships between the coefficients A, C, and D and the ratio C/D.

I understand the wave equations in the three separate regions. For this question I need to only consider I, II. The wave equations need to decrease to zero as x approaches positive or negative infinity. The wave equation and its derivative need to be continuous as well. Thus, the wave equation of I equals II.

My professor did a similar problem last semester, but I can't make sense of his procedure. I think the delta function is in it, etc.

http://i111.photobucket.com/albums/n149/camarolt4z28/6t24.jpg?t=1283054054 [Broken]
 
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  • #2
vela
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Your professor wrote the two equations that correspond to
The wave equation and its derivative need to be continuous as well.
 
  • #3
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Your professor wrote the two equations that correspond to
He did it a bit differently, though.

I also re-worked my part, too.

http://i111.photobucket.com/albums/n149/camarolt4z28/6t24.jpg?t=1283054054 [Broken]

He has

1 = A + B
K = ik (A - B

ik/K = (A + B)/(A - B)

(ik + K)/(ik - K) = A/B = delta

I understand the manipulation up until here. I still don't know how in the heck this helps me related A, C, and D. Do I do the same thing?
 
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  • #4
vela
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I think you're getting your coefficients mixed up. Your equations should be

A = C+D
αA = ik(C-D)

I'm not sure what your professor is doing. It looks like his A and B are your C and D and his K is your alpha. He took (your) A to be equal to 1 for some reason. The delta is not the delta function. It's just the quantity which equals A/B.
 
  • #5
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I think you're getting your coefficients mixed up. Your equations should be

A = C+D
αA = ik(C-D)

I'm not sure what your professor is doing. It looks like his A and B are your C and D and his K is your alpha. He took (your) A to be equal to 1 for some reason. The delta is not the delta function. It's just the quantity which equals A/B.
Oops. I accidentally wrote down B + C for some reason.

Here's what the professor did last week. I assume the book is looking for something like this. I don't know how he got this.

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-08-29000403.jpg?t=1283058414 [Broken]
 
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  • #6
vela
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You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A.

That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.
 
  • #7
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You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A.

That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.
Okay. I'll play around with the equations. Maybe I'll get partial credit. lol.

Oh, for the 1 = A + B, I think he used the free particle solution for the wave heading from the negative x direction towards the potential.
 
  • #8
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You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A.

That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.
Well, I tried to play around with what the professor did, but I couldn't get his equations. Forgive my lack of algebraic-manipulation skills.
 
  • #9
vela
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Rewriting the equations a bit, you get

[tex]\begin{align*}
B - C & = -A + D \\
k_0 B + kC &= k_0 A + kD
\end{align*}
[/tex]

Multiply the first equation by k, add it the second, and solve for B.
 
  • #10
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Rewriting the equations a bit, you get

[tex]\begin{align*}
B - C & = -A + D \\
k_0 B + kC &= k_0 A + kD
\end{align*}
[/tex]

Multiply the first equation by k, add it the second, and solve for B.
Okay. That makes sense. I'm still not sure about my problem. Playing with it, I got

A = [(ik + α)C + (α - ik)D] / 2α

Is that what the book is looking for?
 
  • #11
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Probably not. Try solving for C and D in terms of A. Then you can calculate the ratio C/D.
 
  • #12
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Probably not. Try solving for C and D in terms of A. Then you can calculate the ratio C/D.
Crap. I forgot about the ratio C/D. Well, maybe I'll get partial credit. The homework was due today.

Just now, I got A = [-2ik/(α-ik)] D.

I suspect C would be something similar.
 
  • #13
vela
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I think that's right, and C comes out similarly.
 
  • #14
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I think that's right, and C comes out similarly.
Well, that's just great. I do something right after it's due. I'm not going to do well on this problem set. Pretty much the rest of the problem sets will come from this book. I don't expect to do well on any of them.

https://www.amazon.com/dp/0471057002/?tag=pfamazon01-20
 
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  • #15
vela
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If you don't like that text, you might want to see if your library has Griffith's book on quantum mechanics. I'll admit I've never seen it, but I used his particle physics book as an undergrad. He was very good at explaining concepts and showing how to apply them in problems.

https://www.amazon.com/dp/0131118927/?tag=pfamazon01-20

(It's the top-selling book on quantum mechanics at Amazon.)
 
  • #16
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If you don't like that text, you might want to see if your library has Griffith's book on quantum mechanics. I'll admit I've never seen it, but I used his particle physics book as an undergrad. He was very good at explaining concepts and showing how to apply them in problems.

https://www.amazon.com/dp/0131118927/?tag=pfamazon01-20

(It's the top-selling book on quantum mechanics at Amazon.)
You think it'll help me do the Gasiorowicz problems? Even the professor said Gasiorowicz isn't an ideal textbook. Also, I saw one of my friend's particle physics book, and it's Griffiths, too.
 
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  • #17
vela
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Yes, I think it would help. It couldn't hurt. Textbooks are so expensive, though, so I'd try to look through a copy in a bookstore or at the library first.
 
  • #18
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I have all three Griffiths (unless there are more) books. The EM is best text book of any kind that you will ever find, in my opinion, the particle physics is great, and the quantum book is decent. As for the quantum book: I remember he doesn't even use kets until spin and he still tries to avoid them like the plague, he dumbs down the math formalism too much, and for the most part just tries spoon feed the reader way too much. That being said, he is still as clear and concise as he usually is. Maybe it's good if you don't have a strong math background, particularly in linear algebra and PDEs, but it wasn't good for me.
 
  • #19
vela
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Oh, that's too bad. I forgot I had his E&M book as well as the particle physics book. Both were excellent, so I hoped his QM book would be too.
 

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