Talk:Finite potential barrier (QM)

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I already created an image to explain the notation but think the quality of it could be vastly improved. Maybe the case E < V0 (decaying wave functions in the barrier region) could be incorporated in the same figure, or a separtate figure provided. It would be good if somebody could check the equations and English as well. Bamse 05:37, 31 July 2006 (UTC)

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[edit] Renaming suggestion

I propose renaming of the pages Finite potential barrier (QM), Delta potential barrier (QM), and Delta potential well (QM) to versions without the "(QM)" in the title, as this seems to be the convention used elsewhere such as Finite potential well. Does this seem reasonable?--GregRM 13:48, 24 August 2006 (UTC)

That's fine with me. If the QM is dropped the calculation/explanation for the classical case should be extended though. So far I only wrote a one-line comparison. --Bamse 03:48, 25 August 2006 (UTC)

[edit] What happens when E = V?

I would love to see this explained. It is possible to find a limit for the transmission coefficient as E \to V, but there are apparently no physical solutions to the wave equation. In general however, the E = V case is conspicuously absent from the discussion. 17:43, 23 October 2006 (UTC)

You are right, the case E = V0 is not discussed. I am not sure why you think that there are no physical solutions to the wave equation, though. Let me make two points discussing that limit: (1) Since we have the results for E > V0 and E < V0, and see that the limit E \to V_0 of the transmissions/reflections exists (from above and below the same value), I would not hesitate to use the result in that limit as well. Nature favours graphs that don't have a single missing point, so we could continue the curve at E = V0. (2) As for the calculation. If you put E = V0 in the Schrödinger equation, there are solutions even in the barrier region (linear of the type B1 + B2x ). The rest of the calculation is similar, i.e. matching wave functions and their derivatives in all three regions. The results for t,r are also the same as in the text.
Should we have another calculation for E = V0 in the article? I don't think it is necessary, as it is already implicitly included: expanding the exponentials in \psi_C(x)= B_r e^{i k_1 x} + B_l e^{-ik_1x} to linear order (k1x is small if k_1 a\to 0). Maybe we could add a sentence explaining all this in the text. What do you think? Bamse 03:00, 24 October 2006 (UTC)
Ah thank you, now I see my problem. Indeed, the wave does become linear inside the barrier. However, I don't think that this is implicitly included in the current equation. Yes, in the limit | k1a | < < 1 the solution becomes linear; but when k1 = 0, the solution takes on the form ψC(x) = Br + Bl. And when you lose the non-constant term, you run into my problem. I am writing a program to solve the general piecewise constant problem, and it seems I have to write a special case for k1 = 0. I think the general solution to the DE needs to be treated specially in this case as well. 15:00, 24 October 2006 (UTC)
Yes, it should be treated separately. Basically because the two roots of the characteristic polynomial are the same. I am thinking of adding a section in the article explaining this case. Bamse 07:59, 1 November 2006 (UTC)

[edit] Merge square potential into here

I think square potential should definitely be merged in here. This article is lacking in introduction (it plunges right in to dense math); some of the nice intro from square potential could be used here to give a quick overview so that the casual reader isn't scared off from the start. Also the graphic from square potential complements nicely the one here. (I am a newbie and don't have the confidence to merge articles yet.) HEL 01:52, 28 October 2006 (UTC)

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