Talk:Particle in a box

message: 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)
replies: text: 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)
thread_title:  Degeneracy 

message: In the caption for the energy dispersion diagram, I believe the expression kn=2π/L should read kn=nπ/L.  The reason I say this is that kn 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)
replies: text: 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)
thread_title:  Correct Heading for Dispersion Diagram 

message: 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)
replies: text: 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)
thread_title:  Spatial Location 

message: 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)
thread_title:  Formula for energy levels in higher dimensions 

message: 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)
thread_title:  Particle in a box with boundary at x=-L/2 & x=+L/2 

message: 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)
thread_title:  Time (in)dependence?? 

message: @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)
thread_title:  Periodic particle in a box 

message: 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^\Psi_n|^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)
thread_title:  Essential Self Adjointness and Boundary Conditions