# [Maxima] Bessel_I problem

Richard Hennessy rvh2007 at comcast.net
Mon May 5 08:24:05 CDT 2008

```I guess the same problem goes for the x^6 and x^(2*n) potentials for any integer n.  I was thinking though, the x^(2*n) potentials look a lot like infinite square wells for large n so as an approximation you can get the energy from the Fourier series solution with energy:

E=(%pi^2*hbar^2*n^2)/(2*a^2*m)

where

a = mu*x^(1/4)

This can't be done exactly I am starting to think.  Maybe you can mix together the solution for the x^2 potential with the square well solution and come up with something.  Or I can just live with the recursive formula for the power series coefficients.

Rich

------------Original Message------------
From: Barton Willis <willisb at unk.edu>
To: "Richard Hennessy" <rvh2007 at comcast.net>
Cc: "Maxima List" <maxima at math.utexas.edu>
Date: Mon, May-5-2008 6:45 AM
Subject: Re: [Maxima] Bessel_I problem

-----"Richard Hennessy" <rvh2007 at comcast.net> wrote: -----

>  Can odelin solve Shrodinger's eq for arbitrary Energy for the x^4
potential?

No, odelin will not solve -f'' + x^4 * f = e*f for anything other than
e = 0, I think.  The solutions to -f'' + x^4 * f = e*f are expressible in
terms
of the Heun functions (distinct from Heun's method). I don't have a
proof, but I suspect that e = 0 is the only value for which -f'' + x^4
* f = e*f has a non-Heun function solution.

Odelin knows nothing about the Heun functions. Actually the Heun
function solution to -f'' + x^4 * f = e*f isn't all that useful. Not
enough is known about these functions to determine the eigenvalues
of the x^4 potential in any reasonable way.

> I entered it in for E = 3 and it's still working on it.  Should I let it
run all night?

No, odelin isn't smart about when to stop. Odelin is generating sets
of nonlinear polynomial equations that algsys struggles to solve. Give
your machine a rest. When you stop it, likely it will report something
like

...trying the spherodial wave solver
...solving 21 equations in 7 variables

I don't know about the degrees, but Maxima was trying to solve 21
polynomial (not linear) equations in 7 variables. The spherodial wave
solver almost always generates huge polynomial equations. Maybe odelin
shouldn't try this method unless directed.

Barton
(author of odelin)

```

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