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Re: shooting method

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92024] Re: shooting method
  • From: dh <dh at metrohm.ch>
  • Date: Wed, 17 Sep 2008 04:30:18 -0400 (EDT)
  • References: <gapfd4$o7q$1@smc.vnet.net>


Hi Luca,

there lurk some numerical problems.

First note that the equation is homogeneous, the function is only 

determined up to a constant factor. We may therefore choose the starting 

slope as we please.

As the Potential goes to infinity near zero the function becomes very 

small there. It is therefore numerical difficult to start the 

integration there. It is much less problematic to start the integration 

at the node and integrate towards zero. We therefore set fun[B]==0 and 

arbitrarily fun'[B]==1. And we take care not to set A too far inside the 

classical forbidden region, because this is calling for troubles. Here 

is an example:

A=3/4;B=3/2;

slope=-1;

fun0[E0_]:=Module[{y,x,eq},

eq={-y''[x]+400((1/x)^12-(1/x)^6) y[x]-E0*y[x] ==0,

y[B]==0,y' [B]==slope};

y/.NDSolve[eq,y,{x,A,B}][[1]]]

fun[E0_?NumericQ]:=fun0[E0][A];

res=FindRoot[fun[E0],{E0,-50}]

fun[E0]/.res



Plot[fun0[E0/.res][x],{x,A,B}]

hope this helps, Daniel



Luca Petrone wrote:

> Dear All,

> 

> Using a shooting method, I would like to find the smallest eigenvalue of

> 

> -y''[x] + 400((1/x)^12 - (1/x)^6) y[x] ==  E0*y[x]

> 

> (the quantum-mechanical steady state energy of an anharmonic oscillator with Lennard-Jones potential)

> with the boundary conditions :

> y[A]==0

> y[B]==0

> with A<<1 and B >> 1

> Can anyone help about how to implement it ?

> 

> Thank you

> 

> Luca

> 





-- 



Daniel Huber

Metrohm Ltd.

Oberdorfstr. 68

CH-9100 Herisau

Tel. +41 71 353 8585, Fax +41 71 353 8907

E-Mail:<mailto:dh at metrohm.com>

Internet:<http://www.metrohm.com>




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