Talk:Particle in a box

Comments

User Papa November asked for so comments and feedback.

My primary observation is that this article focuses why to much on explaining how to solve the particle-in-a-box problem in quantum mechanics. Please bear in mind that wikipedia is not a textbooks or a how to guide. As it stands the article reads like a couple of pages from an undergrad physics textbook, not an encyclopedia article. To make the article more encyclopedic I suggest:

1. Less focus on derivation, just state the result with a proper reference to a derivation. Most importantly, less math more prose. You may want to take some space and just explain in plain English what the main ingredients of the solution are. Remember that many of the potential readers of this article don't 'speak' mathematics, so write in English. In any case the solution should probably precide the derivation.
2. The properties of the solution could use some more attention on the other hand. Discrete energy levels and nonzero energy groundstate, are typical for quantum systems. It might be worthwhile to elaborate on that a little. It is important to convey how this problem and its solution related to other problems and the real world.
3. There should be a small history section discussion the who's and when's. Who first proposed the problem, and why? Who first solved it? Did proposing/solving the problem lead to any particular new insights in the development of quantum mechanics?
4. You might want to discuss the role of this problem in the didactics of quantum mechanics. It is one of the first QM problems undergrads learn to solve. This article should 1) mention this. 2)explain why.
5. The article should discuss what kind of practical system can be approximated by this model. What are the applications of this model.

Finally, a word of general advice when writting physics articles. Try not to think too much like a physicist, but try to view the text from the perspective of a layman. (TimothyRias (talk) 09:43, 4 September 2009 (UTC))

I'd prefer more citation and referencing, particularly of the first two introduction paragraphs. I suspect that the reason this article currently reads, quoth Timothy Rias, "like a couple of pages from an undergrad physics textbook", is because much of it has been reproduced from just such a source. I googled a phrase from the intro ("also known as the infinite potential well or the infinite square well") and came up with a number of search results including a page that seemed to be an exam study guide.

My advice is: re-write in an encyclopaedic style and be bold enough to try using your own phrases (instead of just swapping around a word here and there). I appreciate a number of edits have recently gone into this piece but I am a great fan of developing Wikipedia content that has been independently well written as well as prolifically written. A lot of images are cleaned out of Wikipedia, with copy-right cited as reason, and I believe Wikipedians need to be as demanding of their text as they are of pictures in this respect. In short:

• Think about writing Wikipedia content that speaks with flair and originality as well as with an encyclopaedic voice.
• If you are afraid others will question content that you wrote yourself, just be sure to reference it well and cite all of the facts. If someone with more knowledge about the subject, or a sharp eye, reads the article they can easily spot and check or fix a fact that way.
• Think about including sections that might discuss a history of the theory behind the model, or how the model was developed. Has anyone famously discussed the model or based other work on the model? If they have, include this - it is informative and interesting and you can write great readable content based entirely on facts this way.
• Think about including a section that might demonstrate how the model has been applied: give physics a purpose.
• Has any particularly interesting, unusual or notable work in science ever made use of the model? If it has, think about creating either a sub-section or a link to a relevant existing article (or create a whole new article about it!).
• Above all, remember that quality demands bravery as well as accuracy.

I hope I haven't been too harsh a critic. 122.57.176.249 (talk) 07:41, 14 September 2009 (UTC)

Thanks for your comments, but I really must object strongly to your implications of plagiarism. You can look though the edit history and see that I have rewritten all the content in the lead section and the 1D box section from scratch, and not merely swapped odd words from textbooks. The topic is so widely taught, and so standardised in approach that it is an unsurprising coincidence that some phrases have come out similar to some of the hundreds of internet tutorials. As one of the administrators who actually performs the deletion of copyrighted content on Wikipedia, I can assure you that there is absolutely no copyright infringement in this article - just several hours of my original work! If you have specific issues with my writing style, then I'll be pleased to address them, but please don't accuse people of plagiarism without some very firm evidence.

In response to your other comments:

• Can you give some specific examples of unencyclopaedic style, or lack of flair?
• I have provided textbook references throughout the 1D box section to verify everything I have written. Lead sections are just a summary of the main article content, so they don't need to repeat the citations used later in the article. If there is anything you would like a citation for, please could you add a {{fact}} tag next to the questionable sentence in the article, and I will see to it?
• Yes, I agree that a history section would be helpful. As far as I'm aware, it goes back pretty much to the very dawn of quantum mechanics so I guess all the big names in the field have used it for some back-of-the-envelope calculations. I don't have any references that talk about the history of the model, however.
• I can add something about applications of the model, but I only really know about semiconductor devices in any detail. I'll add something about quantum well lasers, maybe.
• It is virtually the only system in quantum mechanics which can be solved by hand, with only a pocket calculator. As such, it is ubiquitous throughout nanoscale science and engineering. I should try to put its importance across more clearly.
• Can you give some specific examples of where cowardice or inaccuracy have affected the article quality?!

Papa November (talk) 09:13, 14 September 2009 (UTC)

I have to strongly disagree with a few of the suggestions made by the IP above. Mainly #1. Removing the derivations is a horrible idea. The subject matter simply cannot be understood without seeing and understanding the derivations. Just because a layman doesn't understand simplistic mathematics doesn't mean that the derivations should be excluded from the article. Physics is spoken in the language of mathematics and it cannot cogently be explained and understood with simple english. Any attempts to do so may satisfy the laymen into mistakenly thinking he understands what the article is about, but would leave the more knowledable reader befuddled with the lack of specifics. As such two people are now confused. One knows they are confused and the other is ignorant of that fact. On the other hand by providing the derivations and hence the actual explanations at least one person will come out after reading the article with a competent understanding of the material. As for who first came up with the notion of a particle in the box... I don't know for sure. But I think it was likely Schrodinger. However, if I'm write that he first solved the particle in the box, its very possible that he wasn't the first to have solved the differential equation of that form.Chhe (talk) 21:19, 24 January 2010 (UTC)

Hi Chhe, please could you clarify a point about your comments? Are you saying that the article (in its current state) needs to have a more detailed derivation or that it's OK as it is? I've tried to strike a balance so that the article is easy for the layman to read and gain some understanding of the topic, while still keeping all the important points of the derivation. Ideally, there should be enough information for mathematically-competent readers to fully understand how the solution was obtained, but not enough to exclude the layman. The important point, I think, is that this is an encyclopaedia article rather than a textbook or a teaching guide. If the advanced reader wants a really rigorous mathematical derivation then we can easily cite a reference which provides this. Papa November (talk) 12:59, 9 February 2010 (UTC)

I reread the current article and from what I can see as before there isn't any derivations in the article. Which is why I don't know what you mean by "more detailed derivations". I don't see any derivations at all in the article. The current article just quotes the results, the wavefunctions and the energies. I want the actual derivations shown. A reader learning this material for the first time would undoubtedly have the question regarding where the wavefunction comes from in the first place. This can easily be accomplished by two methods. Either a section titled "Derivation" can be added or an inline hidden show link can be added that gives the derivation. Whichever is chosen I don't see how either of these will disturb a mathematically ignorant layman's understanding of the article. They will just ignore it. Also, this is overlooking the fact that a layman that doesn't understand basic mathematics won't even be able to understand even the introductory paragraph to the article that is all words. My main concern is for the mathematically versed people who are trying to learn this for the very first time. They will undoubtedly be stumped.Chhe (talk) 02:32, 10 February 2010 (UTC)

I've tried to make sure that an interested reader can easily find every step of the full derivation on Wikipedia. The derivation in the article goes as follows:

• The particle is free inside the box... I provided a link to free particle, where the full 3D derivation is presented but I simply stated the 1D result in this article. The free particle article could be edited to provide the 1D derivation, but I don't see much point in repeating it here.
• The particle cannot exist outside the box i.e. $P(x)=|\psi (x)|^{2}=0$, therefore $\psi (x)=0$ as stated in the article. I have never seen a textbook derivation of the particle-in-a-box problem which really labours this step, but we could add a link to probability amplitude for the interested reader.
• The next step is boundary condition matching. We could expand on this by mentioning the symmetry of the sin and cos components, which provides a sound mathematical analysis while remaining simple to understand.
• Finally, the wavefunction is normalised. I provided a link to the normalisable wave function article which explains the normalisation process.

Is there anything specific that you believe we should add to the derivation? I don't think this article needs to discuss things like the derivation of the free-particle wavefunction, probability amplitudes or normalisation in any depth... that's why we have separate articles for those topics! The only thing that is really missing, in my opinion, is a statement about how the box is "constructed" i.e. the potential $V(x)$ which appears in the Schrödinger equation. Papa November (talk) 10:24, 10 February 2010 (UTC)

The article now actually strikes a pretty good balance between mathematical detail and explanation for the layman. Much better than when I posted the first comments above. Well done! One thing I would add though as an actual mathematical formulation of the problem. That is, Schroedingers equation and the corresponding potential. TimothyRias (talk) 12:20, 10 February 2010 (UTC)

Degeneracy

The section on the three-dimensional case says when two or more of the lengths are the same (e.g. Lx = Ly), there are multiple wavefunctions corresponding to the same total energy. That's true in two dimensions too. The two-dimensional case is discussed first, and doesn't mention this, making it seem as though it first comes up in three dimensions. —Preceding unsigned comment added by 72.75.67.226 (talk) 03:48, 8 October 2009 (UTC)

Well spotted. I haven't checked through the higher dimensional solutions yet. Feel free to correct them! :) Papa November (talk) 08:59, 8 October 2009 (UTC)

Correct Heading for Dispersion Diagram

In the caption for the energy dispersion diagram, I believe the expression k_n=2π/L should read k_n=nπ/L. The reason I say this is that k_n implies "k as a function of n," and yet there is no "n" in the formula. Otherwise this was an article good enough to discuss with my Adv. Inorganic students, and it was nice having access to this information online. Keep up the good work.Rowanw3 (talk) 21:20, 24 November 2009 (UTC)

Thanks very much for pointing out the mistake. I've fixed it on the file description page. In future, please feel free to correct mistakes in articles. Any help you can offer with writing scientific articles would be greatly appreciated :) Papa November (talk) 10:31, 25 November 2009 (UTC)

Spatial Location

The author makes the following assertion:

For the particle in a box, it can be shown that the average position is always <x> = L/2, regardless of the state of the particle.

The above statement is only true if the expectation is taken for an eigenstate. However, the most general state of a particle in a box is a linear combination of eigenstates. For such a general state the expectation value will not always be L/2. See for instance http://en.citizendium.org/wiki/Particle_in_a_box for a graphic demonstration.

PsiStar (talk) 21:13, 2 March 2011 (UTC)

Yes, you're quite right. I phrased it imprecisely... feel free to reword it! I have been meaning to add something about time-evolution of the expectation position in a superposition for a while, but I'm too busy to do it any time soon. The animation on Citizendium is very nice... far more useful to the novice reader than the accompanying equations! Papa November (talk) 00:07, 3 March 2011 (UTC)

Formula for energy levels in higher dimensions

Dear Author, The formula for energy levels in higher dimensions seems to be wrong. For the two dimensional equation it states currently: E_nx,ny = h^2/2m (k_nx + k_ny)^2 = h^2/2m (n_x pi/L_x + n_y pi/L_y)^2. But it should read instead: E_nx,ny = h^2/2m (k_nx^2 + k_ny^2) = h^2/2m ((n_x pi/L_x)^2 + (n_y pi/L_y)^2). As the eigenvalues of the 2D-Laplacian are 2,5,5,8,10 ... = (1+1),(1+4),(4+1),(4+4),(1+9) ... The same applies to the three dimensional equation. Please verify and correct this. — Preceding unsigned comment added by Pia novice (talk • contribs) 23:11, 22 February 2012 (UTC)

Particle in a box with boundary at x=-L/2 & x=+L/2

If the boundary conditions are changed from "x=0 to x=L" to "x=-L/2 & x=+L/2", how to find the two constants A & B. Since in this case none of the constants goes to zero. — Preceding unsigned comment added by Imrohit2611 (talk • contribs) 09:12, 14 April 2018 (UTC)

Time (in)dependence??

The derivation in the article is largely centered around the time-dependent SE, yet a discussion of the discrete stationary states is mixed into the text, without any mention that those are solutions of the time-independent SE. The effect must be thoroughly confusing to anyone who doesn't firmly master Quantum Mechanics but has a serious interest in learning more about it (presumably our target audience).
I would suggest to dedicate an equal amount of space to the time-independent and time-dependent solutions, each under their respective subheadings.
I'm tempted to put a Cleanup template on top based on this issue alone (and there are more on this page).
OneAhead (talk) 14:25, 8 April 2020 (UTC)

Possible original research?

@Luman2009: I'm not a physicist, so I apologize if I'm totally wrong here. Is the material in the new section "A more general analytically solvable problem in quantum mechanics" explicitly supported by the sources? As a layman, it seems to me like it might be original research, so I wanted to double-check. BalinKingOfMoria (talk) 02:11, 22 August 2022 (UTC)

Thank you for asking. Yes, It is original research. The research was published in one article, "Two types of Electronic States in One Dimensional Crystals of Finite Length "（Annals of Physics(N.Y.) Vol 301, p22-30(2002) ), and two editions of the book "Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves" authored by Shang Yuan Ren, published by Springer in (2006) and (2017). Luman2009 (talk) 02:34, 22 August 2022 (UTC)

Sorry. Probably I did not understand your words "original research" correctly. As shown in my previous reply. Those results have been published in a research article and two editions of the same book. Luman2009 (talk) 02:47, 22 August 2022 (UTC)

The mentioned research article and two editions of the book closely focus on the subject and explicitly support the new section. Luman2009 (talk) 02:56, 22 August 2022 (UTC)

Gotcha, thanks! Just for posterity's sake, do you know the relevant page numbers in the article and/or books (so they can be incorporated into the citations)? BalinKingOfMoria (talk) 03:07, 22 August 2022 (UTC)

@Luman2009: Just saw your edit, thanks! The page numbers currently seem really broad--can they be narrowed down to the specific page(s) that say that Bloch waves are a generalization of the particle in a box? BalinKingOfMoria (talk) 03:51, 22 August 2022 (UTC)

Thanks a lot! Actually the whole book focus on the issue and relevant problem as the book title shows. I am also working on a new article "Quantum confinement of Bloch waves" to further discuss the problem.

Probable you can go to see my sandbox if your interested in it. Luman2009 (talk) 04:02, 22 August 2022 (UTC)

(Let's continue this discussion here. I didn't know that noticeboard page existed when I first pinged you here, and in hindsight I should've started the conversation there instead. Sorry for the hassle!) BalinKingOfMoria (talk) 06:07, 22 August 2022 (UTC)

Periodic particle in a box

@Luman2009: Discussion about Kronig-Penney type of model does not belong to this article. This article is about an infinite potential well. If you make multiple copies of infinite potential wells, you do not get any band dispersion, as the electronic states in the wells are completely independent of each other. I have removed the content. Jähmefyysikko (talk) 11:37, 24 July 2023 (UTC)

The above text that has been striken out was based on the misunderstanding of the model. New try: So the model is an infinite potential well with additional priodic potential inside the well. I think it is too far removed from the original model to be discussed here. Jähmefyysikko (talk) 11:41, 24 July 2023 (UTC)

Essential Self Adjointness and Boundary Conditions

The infinite potential well with Dirichlet boundary isn't a physical model, and furthermore fails to present a consistent quantum mechanical system, due to the bad domain of definition of $\Psi (x)$ with the chosen boundary conditions. Some sources:

https://physics.stackexchange.com/questions/362305/whats-the-deal-with-momentum-in-the-infinite-square-well,

https://link.springer.com/book/10.1007/978-1-4614-7116-5 Ch 9.6 "A Counterexample" (ominous),

https://courses.physics.illinois.edu/phys508/fa2017/amaster.pdf Section 4.2.4 (speedy overview), and

https://arxiv.org/abs/quant-ph/0103153 2 "The infinite potential well : paradoxes"

This last paper makes the following observation regarding the Hamiltonian operator and its square:

With state vector $\Psi (x)=-{\sqrt {30}}L^{-5/2}(x^{2}-L^{2}/4)$ (on the interval), we have $H\Psi (x)={\sqrt {30}}L^{-5/2}\hbar ^{2}/m$, and attempts to measure the squared energy yield either:

$\langle E^{2}\rangle _{\Psi }=\sum |b_{n}^{\Psi }|^{2}(E'_{n})^{2}=30\hbar ^{4}/(m^{2}L^{4})$

where $E'_{n}$ are even energy eigenvalues, versus

$\langle E^{2}\rangle _{\Psi }=\langle \Psi |H^{2}\Psi \rangle =\langle \Psi |0\rangle =0$. Disaster.

It all comes down to the spectral theory: The construction given with boundary conditions $\Psi (\pm L/2)=0$ is a space where the momentum operator $-i\hbar \partial _{x}$ fails to be an essentially self-adjoint operator, after this everything falls apart. One can only resolve this by modifying the wave function space, with twisted periodic boundary conditions $\Psi (-L/2)=e^{i\theta }\Psi (-L/2)$ for some $\theta$ to be decided. The intuition that $\lim _{x\to L/2}\Psi (x)=0$ is wrong, and perhaps fails us here because the potential is infinite. A better intuition is that our wave functions should start in $L^{2}(\mathbb {C} )$, and position eigenvectors may end up being continuous under constraint by the system.

Of course, this particular particle-in-a-box is still a useful model to study in the beginning, which creates a tension. But surely the fact that the derivation leads to contradictions, enforces bad intuitions, obscures the free variable $\theta$ and the physically relevant twisted boundary, and is fundamentally incompatible with the underlying theory should be signposted somewhere for readers. There seem to be a lot of misconceptions out there regarding these questions, and this is a fundamental example in the pedagogy. Jagmanjg (talk) 09:36, 19 March 2024 (UTC)