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#1 2017-01-19 07:15:07

art1915
Member
Registered: 2017-01-19
Posts: 2

2nd order differential equation from Quantum Mechanics

Related to a physical problem in Quantum Mechanics the following differential equation arises:

(l (l + 1) + b (x^3) ((x^(3/4))/((-1 + x)^(7/4))) - 2 b (x^2)) y - 2 x y' + x (-1 + x) y'' == 0

with the border conditions:

Limit[y[x], x -> 1] = 0

and

Limit[y[x], x -> Infinity] = 0

and we can set y[x]=0 for all x<1 (that means, for x<1 a solution other than the trivial solution y[x]=0 makes no physical sense)

The term l(l+1) corresponds to an angular momentum with l=...-3,-2,-1,0,1,2,3...

And b is an energy related Eigenvalue, perhaps depending on l and additional integer numbers (quantum numbers).

This differential equation has an irregular singularity at x=1 . That means, there is no standard method, for example like the Frobenius method, to solve this equation, because the solution will provide terms, which can not be developed in a Taylor series at x=1. A term with this property would be for example E^(-1/(x-1)).

The problem of this differential equation for me is, that I could not find any transformation to a known and solved differential equation. The numerical solution shows a shape of the function as expected. The ground state will provide a function starting from y[1]=0 (boundary condition) rising to a maximum and will finally fall down to 0 (limit to Infinity). I can imagine the excited states. The first excited state will show a similar behaviour at x=1 and infinity, but the function will cross the x-axis. The second excited state will cross the x-axis two times and so on.

This differential equation describes matter in a new kind of understanding. I am convinced, that this problem will be solved by intuition. Mathematica and Wolfram Alpha failed. Solving this differential equation means, to find the right trick. For this I give it a try in this forum. Perhaps there is someone who is able "to see" the right way.

Regards

Robert

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#2 2017-01-19 09:34:35

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 2nd order differential equation from Quantum Mechanics

Hi Robert;

I tried Mathematica 11 on it and it just ran until I gave up. The odds are that it does not have an analytical form. I do not know of a substitution or anything else that can deal with it. You can take it over to Math.SE maybe they can help.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2017-01-19 22:01:33

art1915
Member
Registered: 2017-01-19
Posts: 2

Re: 2nd order differential equation from Quantum Mechanics

Dear bobbym,

thank you very much for your help. What is Math.SE?
Do you mean http://math.stackexchange.com/ ?
If so, I will give it a try.

The best would be to collect all my investigations into one large mathematica Notebook file. I did not want to provide it, because it could influence the helper in a bad way perhaps.

Regards,

Robert

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#4 2017-01-20 00:53:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 2nd order differential equation from Quantum Mechanics

That is the SE I was redferring too. What version of Mathematica are you using? Keeping good notes is mandatory and .nb's are the best way to do that.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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