Wikipedia:Reference desk/Archives/Mathematics/2007 October 22

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[edit] October 22

[edit] Numerical Solution of Schrödinger Equation in One Dimension?

I was wondering if there was a way to solve the Schrödinger equation numerically with a user-supplied potential function on a certain interval. I know about the parts that match normal solution of differential equations, but I was wondering how to set boundary conditions and how to numerically find a valid value of the energy of a bound state. --Zemylat 17:25, 22 October 2007 (UTC)

From experience except in the simplest cases the differential equation is not analytically solvable.
I was wondering what you meant by 'set boundary conditions' - did you mean ways to create energy boundaries in V(x)? (This can easily be done with equations similar to V(x) = fn(exp((x-a)n)) where n is large odd integer eg 1, 'a' is the position of the boundary ( x must be less than a ) and f(x) can be exp(x2) or x2n or just x etc .. for boundaries such as a<x<b use V(x)=fn(exp(a-x)2n) +fn(exp(x-b)2n) etc . An alternative energy barrier for x<a is V(x) = fn( (x/a)2n )
f(x) is there just to make the energy barrier 'more sharp' - without having to resort to discontinuous functions....
The energy of a state comes easily if you can solve the differential - I'm sure it's covered at that page.
The differential might be solvable for a specific case of V(x) - if you have that and are stuck on it someone (probably not me!) might be able to help you.. Otherwise you need to ask about getting approximate solutions to differential equations...87.102.83.3 20:40, 22 October 2007 (UTC)
Yes, I'm talking about a numerical solution, not an analytic solution. I remember a piece of software a while back called MP Desktop had a numerical solver for Schrödinger's equation, and from what I could tell, it integrated from either end toward the middle, and if the energy value given wasn't a valid bound state, the wave function would be discontinuous. But I don't know exactly how that worked. --Zemylat 14:07, 23 October 2007 (UTC)
So the software checked if a particular function was a valid solution of the equation? (Or did it work the other way round and try to find a wavefunction for a particular energy).. The second way is much more difficult..
Your original question asks how to get the energy - for this you need a wavefunction - if you can give more details - such as any values/function/potential energys you want to include - that would help a lot.87.102.7.135 18:23, 23 October 2007 (UTC)
It just occurred to me that what you where discribing was the Particle in a box problem see also (http://ww.google.co.uk/search?hl=en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=particle+box+schrodinger&spell=1 first few links) - this is a special case - in fact this problem is more like the equation of motion of a vibrating string.. In this case if the box edges (boundaries) are at A and B then the wavefunction at A and B is zero. (technically this specific problem isn't really suited to modelling with the schroedinger equation since the function is discontinuous - and there are various problems associated with that..)87.102.10.72 22:07, 23 October 2007 (UTC)
What in the world do you mean "this specific problem isn't really suited to modelling with the schroedinger equation"? Solving the Schrödinger equation for the particle in an infinite square well is one of the first exercises in any quantum mechanics course. —Keenan Pepper 03:37, 25 October 2007 (UTC)
Have you looked at the WKB approximation? —Keenan Pepper 03:10, 24 October 2007 (UTC)
Isn't that almost totally useless after (in the text) the point where it says "Next, the semiclassical approximation is invoked" - since it basically 'bodges' the solution by ignoring all the terms in the series except A0, and ignores the differentials etc etc87.102.94.157 12:19, 24 October 2007 (UTC)
No, the WKB approximation is not "almost totally useless", and it doesn't "bodge" anything. It's a perfectly valid limiting approximation for a large class of potentials. You think you're smarter than Wentzel, Kramers, and Brillouin? —Keenan Pepper 03:32, 25 October 2007 (UTC)

[edit] wiki project in mathmatics

my son has a project and it is about a project of the year which is math and needs to know something about the number 20 and its vocabulary i guess that is what it is. please help if you can. he is in 5th grade —Preceding unsigned comment added by 76.182.221.170 (talk) 22:20, 22 October 2007 (UTC)

We have the article 20 (number). PrimeHunter 22:33, 22 October 2007 (UTC)