Talk:Rectangular potential barrier

Untitled

I already created an image to explain the notation but think the quality of it could be vastly improved. Maybe the case $E<V_{0}$ (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)

E<V

I noticed the only representation of this article was for the free states, but would it not be insightful to also explain the bound(tunneling) states as well?128.235.8.144 (talk) 20:52, 6 June 2017 (UTC)

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)

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 $E=V_{0}$ 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 $E>V_{0}$ and $E<V_{0}$, 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 $E=V_{0}$. (2) As for the calculation. If you put $E=V_{0}$ in the Schrödinger equation, there are solutions even in the barrier region (linear of the type $B_{1}+B_{2}x$ ). 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 $E=V_{0}$ in the article? I don't think it is necessary, as it is already implicitly included: expanding the exponentials in $\psi _{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}$ to linear order ($k_{1}x$ 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 $|k_{1}a|<<1$ the solution becomes linear; but when $k_{1}=0$, the solution takes on the form $\psi _{C}(x)=B_{r}+B_{l}$. 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 $k_{1}=0$. 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)

It would be great to add a note, where the linearity of the $E=V$ case comes from. I was just doing my QM homework and without this discussion page I would have been struggling a lot. — Preceding unsigned comment added by 88.76.159.136 (talk) 12:39, 31 October 2011 (UTC)

The expression for transmission for E=V0 case has slight problem. The formula is for the barrier f length 2a while in the article the length considered is always a. I will be happy if someone rechecks the calculation.

Also, I believe that the E=V0 case should be done separately, since the transmission and reflection coefficients are different in this case.

I agree the expression was incorrect as written previously; it applied to the case of a well of twice the length. 82.44.207.216 (talk) 17:23, 22 August 2020 (UTC)

I added a derivation for the transmission coeffficient for the case $E=V_{0}$ as well as a citation for the expression itself. I did not add this derivation in the mathematical derivation section because there no expression for $T$ is stated.Galaktico (talk) 13:44, 3 July 2023 (UTC)

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)

Done, at 06:27, 9 November 2006. This is the last version of the square potential article prior to the merge, and here is the content that was merged. The only merging of the term square was the description of the image (right)

. As a non-expert, I'm left wondering, what is the difference between a rectangle and a square (potential)? Is the square potential a special case of the rectangular potential? Wbm1058 (talk) 17:57, 8 December 2011 (UTC)

value for k1

It seems to me that $k_{1}$, if you assume the wave function for region B is of the form $B_{r}e^{ik_{1}x}$, should be ${\sqrt {2m(E-V_{0})/\hbar ^{2}}}\quad 0<x<a$. That is the value you get for $k_{1}$ if you plug the wave function back into shrodingers. Also it's actually consistent with the statement that if E is less than $V_{0}$ the term becomes imaginary and the wave function becomes an exponential decay. Correct me if I'm wrong. Melink14 04:09, 10 April 2007 (UTC)

I agree. I changed it. --69.157.68.128 01:34, 13 April 2007 (UTC)

Transmission probability of a finite potential barrier figure

This figure says that {\displaystyle {\sqrt {2mV_{0}}}a/\hbar =7} {\sqrt {2mV_{0}}}a/\hbar =7. According to my calculations, this is wrong and could be around 47. — Preceding unsigned comment added by 83.248.116.80 (talk) 21:18, 17 September 2017 (UTC)

Redirect

Potential barrier redirects to this article. However, this article is not about potential barriers, but instead about one specific potential barrier. I believe that a stub about a general potential barrier, and how it comes about from classical potential energy would serve better for this specific search. Brent Perreault (talk) 13:15, 31 August 2010 (UTC) Wbm1058 (talk) 17:19, 8 December 2011 (UTC)

the width of potential goes to the infinite ?

If the width of potential goes to the infinite, this belongs to step potential problem ? barrier potential problem ?
The textbook doesn't deal with the case when the width of potential goes to the infinite.
I find out that the rate of reflction doesn't converge to that of step potential when the width of potential goes to the infinite.
How do you think of this case?
If anyone has the interest in this case, contact me, mrhun@hanmail.net
— Preceding unsigned comment added by Younghun park (talk • contribs) 01:50, 6 July 2011 (UTC)

I am not sure whether taking the limit of infinite barrier width makes any sense. Note that the derivation assumes that there are regions to the left and right of the barrier with the same wavenumber $k_0$. Taking the limit of infinite width you are not going to get rid of the region beyond the barrier and should not necessarily obtain the same results as for the step potential. bamse (talk) 21:25, 7 June 2017 (UTC)

Is this the only type of "potential barrier" that exists? Should there be a general "potential barrier" page, or is that too abstract?

I searched on Wikipedia for "potential barrier" and it redirected me here. This seems like a very specific type of "potential barrier", rather than the entirety of the subject. But I could be wrong, as my physics is a bit rusty.

Transmission probability

Is it correct to take C(L) = 0 to ensure that no particle is incoming from the right? If you calculate the square of the module of psi on that assumption (C(L) = 0), you will easily notice that it is constant on the right side of the barrier while oscillating on its left side. Such a difference seems very strange to me.

In my opinion, it is better to take B(R) or B(L) = 0 — the one that stands before the raising exponential function. This leads to the picture shown here — oscillation, followed by exponential decay, then oscillation again — which seems more appropriate.