Talk:Particle in a box
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Thanks! A nice, detailed article.
Maybe we should mention that we are using the time-independent Schrödinger equation here, which is different from the one on Schrodinger wave equation. Or maybe that page should be changed.
Also, is it a coincidence that all ψn are real-valued functions? Shouldn't complex numbers show up? --AxelBoldt
- The eigenfunctions of the time independent Schrodinger equation can always be made real. Phys 16:01, 17 Aug 2004 (UTC)
Thanks for the compliment on the article. I just added a statement that we are using the time-independent Schroödinger equation. I think it would be useful to add this equation to the Schrodinger wave equation page and explain when its use is appropriate (probably a task for someone more knowledgeable on this subject than myself).
As for the absence of complex numbers, my understanding is that complex numbers, indicate the "phase" of the system and arise in the time-dependent equation. Since the particle in a box is a time-independent problem and corresponds to a standing wave there is no time-dependent portion and hence no complex component to the solution. If anyone more knowledgeable than myself on this subject can confirm or deny this explanation, I would greatly appreciate it.--Matt Stoker
About the doubt above: the wavefunctions for time-independent quantum mechanics problems ARE complex in general, and not necessarily real. The example in this article is simple and compelling because they are real, but in general they can be complex. A simple example of a complex standing wave are the electron states in a hidrogen atom. - Ernesto.
In this case, the wavefunctions are real because the problem is one-dimensional. Generically, one-dimensional solutions to the S. wave equation have no phase variation, so we might as well take them to be real. -- CYD
The math markup on this article is ghastly. The hbar looks horrible, the fractions are done using (num)/(den) which also looks horrible when it is done everywhere, and the integral limits look horrible as well. I would like to Tex-ify this article when I have time, or someone else can do it, unless there is large protest against it. --dave
As for: I would like to Tex-ify this article
- The (fortunately!) few instances of TeX-urication in Wikipedia seem to make the fonts of their articles ghastly inconsistent, both in size and in type.
- I've seen this complaint before, but I don't really buy it. The fonts for the formulas are bigger than the text, but the formulas are all the same size. So I don't really see any "inconsistency". The font size of the formulas are just bigger than the text, which makes them stand out. It's no big deal if the formula stands out. For the tex-math stuff inline, this can screw up the spacing between lines, but but only for certain math things. Just simple greek symbols don't cause any problem.
As for: The hbar looks horrible
- How does one set one's Wikipedia:Preferences for how HTML operators (such as "<strike>") are implemented and rendered? And: why might this article require any particular "struck-out unit" anyways?
- Best regards, Frank W ~@) R 20:34 Apr 2, 2003 (UTC)
- I'm not sure if I understand you, but the "struck-out" unit is necessary in this case to adhear to convention. It is h-bar which is a standard unit in quantum mechanics. It prevents having to write out 2*Pi everywhere. --dave
Ok, about the above stuff....I think tex for the indented formulas is great. Much better than the html. But tex is definitely bad for inline stuff. But I think using html in the inline stuff is ok, and yes that makes the font size inconsistent of variables with the main formula, but it's okay, it's still the same letters and it can still be italic, so I don't see this as a huge factor. At least, I think having the main formula in TeX looks so much nicer that it outweighs any concern for the font sizes being inconsistent. --dave
Does anyone know why formula in 18 and 19 the square root sign is slanted in one, but not the other? Thanks. --dave
- pure speculaion from my part: it's because 'y' is slighty longer than 'x'; if you replace:
- <math>Y_{n_y}=\sqrt{\frac{2}{L_y} \sin \left( \frac{n_y \pi y}{L_y} \right) } \quad (19)<math>
- by
- <math>Y_{n_y}=\sqrt{\frac{2}{L_y} \sin \left( \frac{n_y \pi y}{L_x} \right) } \quad (19)<math>
- ^
- |
- then both are slanted! -- looxix 20:30 Apr 3, 2003 (UTC)
- Thanks looxix, I tried fixing it using \vphantom{y} but it doesn't support that command for some reason.... --dave
"Particle in a box" analysis is said to be invalid for certain shapes.
Chemists, led by graduate student Heng Yu and William E. Buhro, Ph. D., professor of chemistry in Arts & Science, at Washington University in St. Louis have published an article in Nature Materials, August 2003,which is described in a press release as "the first comprehensive comparison relating shape to the phenomenon known as "quantum confinement"
http://www.sciencedaily.com/releases/2003/10/031014071540.htm
The release states: "After compiling all of the data graphically, the group found that the famous "particle-in-a-box" calculation -- a quantum "model" both simple enough for college freshmen and rigorous enough to explain observed quantum phenomena – fell short of predicting absolute values for individual shapes , due to its simplification of the nature of diameter dependence.
"However, they were encouraged to discover that particle-in-a-box approximated the ratio of the slopes of wires to dots, a fact verified by their experimental and theoretical data. This is a particularly advantageous finding because it gives other researchers a method of assessing 2-D confinement in their own quantum wires without the need of a supercomputer."
Ken A.
Scienceworld Links
For one, I think that link to wolfram should be removed. Anyone look at that page?
- I think I fixed it. Wikipedia should have its own finite well page. PAR 17:48, 1 Apr 2005 (UTC)