Talk:Particle in a box

Do we need the 3D case?

I think this page is starting to approach complete, besides the currently empty sections on the 3D spherical and cubic wells. I believe that the cubic will isn't really needed, but what are people's thoughts on the spherical well? It is definitely important but perhaps a separate page for it would serve to keep this page simpler, as well as making it nearly done.

Michael Underwood 20:50, 4 July 2007 (CDT)

The simplest 3D case is a cube, which is worth treating here. The ball case is an exercise in spherical coordinates, maybe better suited for a different article. What I would like to do here is to make an animation of the probability density of a simple non-stationary state. /Pieter Kuiper 04:13, 23 October 2007 (CDT)

I agree, I was getting ready to move the spherical well to its own page anyway and have now done so. Michael Underwood 14:31, 23 October 2007 (CDT)

Excellent. I made the animation that I was thinking of, and I put it in below your image, but that is probably not the best place. Of course one should write an explanation, but I do not have the time now. /Pieter Kuiper 17:36, 23 October 2007 (CDT)

Readability

Not sure what the 'accessibility' test is for maths articles so I apologise if the following comments seem ridiculously simple and silly - I did A-level pure and applied maths 20 odd years ago, but that's when I said goodbye to calculus and 'hard sums'. There's a few instances of acronyms that aren't explained or linked to which I found made the article presuppose quite a bit of knowledge, nothing too testing - I put (1D) in brackers after one-dimensional to aid reading for non-mathematicians such as myself. Is an ODE some kind of differential equation? perhaps we could spell it out in the first instance and contract it for later instances? --Russ McGinn 17:44, 23 October 2007 (CDT)

Russ, ODE is indeed some kind of DE. It stands for ordinary differential equation. Thanks for pointing that out; I've changed it in the text. You mention "a few instances" - do you have any other input on how the article reads in general or sections that aren't as clear as they could be, or did you already list them all? Michael Underwood 18:00, 23 October 2007 (CDT)

Sorry Michael, I think I may have overstated 'the instances'. I'm not sure I'm really your man to comment on whether sections are as clear as they could be. Quantum mechanics, like relativity, is something to me where I was taught the basic ideas in school, love the wierd and wonderful 'effects' we are told about, but am instantly lost in the detail :-) I hadn't realised this was a sub article of Schrödinger equation a read of which helped, but you've already linked that in the first sentence so I can't see how that can be improved. I was staring at the first equation wondering what psi represented, but the other article tells me it's the eigenvector or wavefunction or (as I think I was taught) a quantum value. I'm at the limit of memory on the d2/dx2 bit too - that's the differential calculus bit I think........but maybe I get the general idea - within certain fixed values in 1D space the particle will have variable potential and because of that potential the particle cannot move beyond the fixed limits in space? Actually I've just realised I've no idea what 'potential' means in this context, so I'm off to read Quantum mechanics and Schrödinger equation rather than waste your time with silly questions. cheers --Russ McGinn 19:46, 23 October 2007 (CDT)

No, these are excellent questions. I feel that one should strive to make these articles as accessible as possible. A wide audience should be able to read the lead and the introduction. The term "potential well" is a bit of a conceptual hurdle, as there is no potential inside the box: the particle is locked in between two impenetrable walls. Classically, it is bouncing back and forth at arbitrary speeds. Quantummechanically, it is a standing wave in the probability density wave function, with quantized kinetic energies. /Pieter Kuiper 01:34, 24 October 2007 (CDT)

Two remarks

This article is a very good start of the CZ career of Pieter Kuiper. Pieter, I hope you will add many more of this caliber. I don't have any consolation for Russ, the times are long gone that informed laymen could follow science (I wonder if Sir Christopher Wren could understand Sir Isaac Newton, may be just so, but soon after this changed completely.) So Russ, you have to live with it, just as I don't understand Heidegger and I know it.

My remarks are:

1. The proper classical equivalent of a particle in a box is a particle with a given initial position and non-zero momentum. (The drawing implies zero momentum). With constant kinetic energy the particle will move through the box as a pool ball on a frictionless pool table. Collisions with the walls are elastic (no energy absorbed by the walls), so the pool ball will forever bounce back and forth on the table.
2. The second panel of the second figure is nice and requires the following explanation: If the system is initially (i.e., at time zero) in a state ψ = sin x + sin 2x, then we must use the time-dependent Schrödinger equation to find ψ at later times. (The time-independent SE may be used only if the system is initially in eigenstate of H). Solution of the time-dependent SE equation gives (with hbar = 1)

$\psi (x,t)=e^{-iE_{1}t}\sin(x)+e^{-iE_{2}t}\sin(2x).$

Use (in appropriate units)

$E_{1}={\frac {1}{2}}\quad E_{2}=2.$

Then

${\begin{aligned}\psi (x,t)\psi (x,t)^{*}&=[e^{-it/2}\sin(x)+e^{-2it}\sin(2x)][e^{it/2}\sin(x)+e^{2it}\sin(2x)]\\&=\sin ^{2}(x)+\sin ^{2}(2x)+2\cos(3t/2)\sin(x)\sin(2x)\,\end{aligned}}$

and the very last function is visualized in the second panel of Fig. 2 (at least I would bet that this is it) as a function of t.

Thanks for your compliment! I am planning some kind of animation that will show the rotating phases of the eigenstates. And I agree that the ball in the first figure is likely to be misunderstood as lying still. A velocity arrow might help, but I will see if I can produce something a bit more dynamic without resorting to an animation. /Pieter Kuiper 09:45, 25 October 2007 (CDT)

Based on these comments I've updated the image I originally included to show the particle with non-zero velocity. I would like to point out though that classically the particle is perfectly allowed to have zero kinetic energy. Also, while a stationary classical particle doesn't correspond directly to a quantum one with non-zero kinetic energy it might for some be the most intuitive view of a stationary state. It's nice to finally have some discussion about this article; for the first five months that I worked on it I didn't get any feedback and now several people are contributing, thanks! Michael Underwood 14:13, 25 October 2007 (CDT)

I am sorry Michael, I overlooked your article, and sincerely thought that it was written solely by Pieter. Your comment triggered me to have a look at the history and now I see that you already wrote most of it quite a while ago. Mea culpa.

As you know well a quantum mechanical particle cannot be "at rest somewhere" (with well-defined position and zero momentum). That is why I suggested that a moving classical particle would be a better analogy.

To Pieter I like to say that I strongly believe that his nice (and illuminating) picture needs an explanation along the lines I gave above. Otherwise it is just a moving picture like so many on the Internet. The explanation above is based on the very important fact that the time-dependent Schrödinger equation is the fundamental equation of quantum mechanics. It is not Hψ = E ψ —the equation displayed on so many a T-shirt at physics conferences.--Paul Wormer 03:12, 26 October 2007 (CDT)

Paul, not to worry, I'm just glad you are happy with the article so far. The main part that Pieter did in fact add was the animation, which I have now moved to a new section about non-stationary states and explained in a manner similar to what you gave above.

Pieter, I think it would be great if you added the $x=0$ and $x=L$ labels on the image, and perhaps labelled at least the vertical axis. What do you think? Michael Underwood 15:34, 26 October 2007 (CDT)

The figure does not really match the text, but I had not really intended to write a text that is so mathematical. I made the drawing as general as possible, to help explain things in a more conceptual or handwaving way. The black lines are not just axes, they also allude to a box or rather a container, in which water is sloshing back and forth. Putting a label on the vertical axis would destroy the ambiguity. I prefer explanations in words and images. /Pieter Kuiper 15:56, 26 October 2007 (CDT)

I'm not sure what you mean that the figure doesn't match the text. I animated the probability density function calculated in the text and it appears to be identical in shape to the one in your figure. Do you think that we need a different animation, or a different initial state for the particle? As for adding the labels to the axes, removing ambiguity is exactly why I thought it should be done... Michael Underwood 16:36, 26 October 2007 (CDT)

One thing that makes this sort of article much more accessible is to give examples of how this may be applied to real-world problems - this is a comment coming from an adult educationalist. Lacking this, nonspecialist will read it and say, "Okay...so what!?"  :-) Stephen Ewen 03:41, 26 October 2007 (CDT)

• Hi Stephen, quantum mechanics and "real-world" is almost a contradictio in terminis. Real-world is the world around us of very heavy particles (even the tiniest dust particle is enormously heavy compared to an electron). It took (and takes) physicists lots of mathematics and mental power to get quantum mechanical pictures right, because QM is so far removed from our daily life experiences. You know that Feynman once said: "somebody who claims to understand quantum mechanics, hasn't understood quantum mechanics". Further, what for a physicist would be a "real-world" application of a particle in a box, for instance Quantum Corrals, would for a non-physicist still not be very clarifying. --Paul Wormer 04:36, 26 October 2007 (CDT)

PS. I can live with the OK, so-what attitude of non-scientists, I lived my whole life with it. But don't forget the technology that QM brought you: transistors, LEDs, MRI scans, laser printers, compact and hard discs, computer hardware and so on and so forth. Without QM we couldn't have had this discussion. --Paul Wormer 04:36, 26 October 2007 (CDT)

Well, you've just answered how this may be applied to real-world problems, haven't you? A better way to scan the body, a better way to store data, etc.  :-) Stephen Ewen 04:46, 26 October 2007 (CDT)

Stephen, Paul did list some of the many things that quantum mechanics as a whole is responsible for... It's a little harder to point to those examples and say where they make use of the solution to the problem of a particle in a one-dimensional infinite potential well. This problem's true contribution to the 'real world' is most likely just in the fact that people who developed things such as lasers and MRIs probably started learning QM with an example similar to this one. If they'd never learnt it they likely couldn't have gone on to develop what they did, so its use is primarily pedagogical I suppose. Michael Underwood 15:34, 26 October 2007 (CDT)

Retraction

I retract what I wrote above about motion on a pool table. This is true for a particle in a 2-dimensional box, whereas here we are talking about a 1-dimensional box (motion on a line). The word box and the 2-dimensional picture (with energy on the vertical axis and position on the horizontal axis) put me on the wrong track, but this is no excuse, I should have known better. To put the record straight: classically the particle moves back and forth on a line with constant speed and this speed can have any non-negative real value (is not quantized). --Paul Wormer 04:10, 26 October 2007 (CDT)

Yes, the word "box" and the box-like picture tend to be confusing. That is why I tried to draw a particle on a line. I cannot do an upload here now, but I put it here on Commons. It is not ready yet - I want to show the ball bouncing between redbrick walls, and I am still trying to get the svg pattern syntax right. /Pieter Kuiper 09:07, 26 October 2007 (CDT)

What do people think of the current image I have at the top of the article? It shows motion, in one dimension only, and clearly labels that the vertical axis is energy and not a second position variable. I don't think it's any less illuminating than what Pieter is hoping to produce, but perhaps I'm biased... What word would be better than the two used now (box and well)? They are the ones I've always encountered for this problem while studying QM. Michael Underwood 15:33, 26 October 2007 (CDT)

Certainly. I stopped working on my image when Michael had uploaded his new version with the motion trail and the velocity arrow. I think it is a good picture, it also explains the potential, and it should be in this article. /Pieter Kuiper 02:17, 27 October 2007 (CDT)

Tutorials?

I recently noticed that there is a (proposed?) subpage type for Tutorials. I wonder if the bulk of this article is in fact more suited to being a tutorial, with only the introductory blurb describing what one would find if they ventured to the tutorial? But then it also seems that in physics a rather large number of articles might end up being tutorials attached to stubs.

Perhaps the best option would be to transfer most of the article to a tutorial, perhaps bulking up some of the derivations slightly, and leave only the results in the main article. Something along the lines of "The potential is V(x)=... which in the SWE admits the solutions En=... and ψn=...".

Does anybody know of any examples of the tutorial subpage currently in use, or know anything about the status of tutorials on CZ in general? Michael Underwood 15:42, 26 October 2007 (CDT)

That sounds like a good idea. Many science articles have the style of a tutorial, also on Wikipedia (but there such articles are often not sufficiently coherent). One could have encyclopedia-style articles, like what Russ McGinn was looking for, and parallel tutorial articles with TeX-formatted derivations. CZ:Subpages says that tutorials are supposed to be "simplified presentations of the topic" - I do not know if that is the main meaning of the english word.

Tutorials: one or more pages that introduce a topic specifically for students; would be focused on more "practical" aspects of the topic, have more examples, and even perhaps some problems at the bottom of the page; some topics may lend themselves to more explicit "how to" type instructions (how to change your oil); main content may or may not exist on the "Tutorials" page itself.

I am very new here, but I could not find anything more about it. Maybe this is something to discuss on the tutorial talk page? /Pieter Kuiper 03:09, 27 October 2007 (CDT)

Including radiation

"If the particle is charged, the oscillating charge density will produce electromagnetic radiation (light) with photon energy equal to the energy difference between the levels n = 1 and n = 2."

I know what is meant here, but I'm not sure if I like off-handedly throwing it into the discussion as the only mention of charge and radiation. Combined with the periodicity of the function and animation, it gives the impression that light will be continuously streaming out of the particle (or at least that once a period a photon will emerge). There's no discussion of where the photon will go, how the particle's state will change afterwards, how the photon will interact with the potential, that it's likely to be re-absorbed by the particle if it can't penetrate the barrier... I think we could do better with a link to cavity QED or quantum optics or something, instead of trying to say all this in the particle-in-a-box article. Comments? Michael Underwood 17:06, 26 October 2007 (CDT)

This is not cavity QED - it is the particle that is in the box, not the radiation.

There are probably better places than this article to provide further explanation, like quantum jump (to name a catchy title). But I am a slow writer, finding it easier to point out a consequence in an off-hand sentence. /Pieter Kuiper 03:19, 27 October 2007 (CDT)

Phases.

Let me first say that I agree with Michael on several accounts. Of course, the particle-in-a-box-problem per se did not lead to MRI, but QM did this, interwoven with all its concepts (for MRI most importantly spin and its quantized energy in a magnetic field).

Since Pieter's animation has a clear mathematical representation, we should not hide that fact and not use handwaving instead; on that point I agree again with Michael. In my opinion equations come first and handwaving arguments second. That is not to say that I deny the power of the latter. They can guide us to conclusions (which then must be proven mathematically to be correct), and they also help one in remembering certain facts. But once a theory is mathematically well-founded, only handwaving can easily give a misguided impression of a theory. In my field (theoretical chemistry) Coulson's qualitative book Valence did lots of harm. It made whole generations of chemists believe that they could do QM without a firm base in mathematics. I've met several physical chemistry professors with completely misguided views of QM that they derived from that book. (Of course Coulson himself knew QM perfectly).

With regard to phases: maybe we should work out the non-stationary exercise in a little more detail in the article (instead of having a "tutorial page", CZ is not a textbook, is it?).

Using hbar = 1, L = π, and mass m = 1 the TD-SE reads

$-{\frac {1}{2}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}=i{\frac {\partial \Psi (x,t)}{\partial t}}$

It is not difficult, given the initial expansion in (unnormalized) eigenstates of H,

$\Psi (x,0)=\sin(x)+\sin(2x)\,$

to show that

$\Psi (x,t)=e^{-iE_{1}t}\sin(x)+e^{-iE_{2}t}\sin(2x)$

To determine E₁ and E₂ we plug this back into the TD-SE and use

$-{\frac {1}{2}}{\frac {d^{2}\sin(kx)}{dx^{2}}}={\frac {k^{2}}{2}}$

and

$i{\frac {d}{dt}}e^{-iE_{k}t}=E_{k}e^{-iE_{k}t}.$

Hence

${\frac {1}{2}}\sin(x)e^{-iE_{1}t}+2\sin(2x)e^{-iE_{2}t}=E_{1}\sin(x)e^{-iE_{1}t}+E_{2}\sin(2x)e^{-iE_{2}t},$

from which

$E_{1}={\frac {1}{2}},\qquad E_{2}=2.$

It makes sense that the energies are positive, because they are pure kinetic energies, which also in QM are positive. So, the total unnormalized wave function is

$\Psi (x,t)=e^{-it/2}\sin(x)+e^{-i2t}\sin(2x),\,$

which is slightly different from the result given by Michael. The difference has no influence on Pieter's animation. --Paul Wormer 03:59, 27 October 2007 (CDT)