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)

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)

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)