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)

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

Then

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

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.