Computational Physics

Exercises 27

Exercises

2.1 [C] Try using the Runge–Kutta method with an adaptive time step to integrate the
radial Schrödinger equation in the program ofSection 2.2, keeping the estimated
error fixed as described inAppendixA 7. 1 .What is the advantage of this method over
Numerov’s method for this particular case?
2.2 [C] Consider two radial potentialsV 1 andV 2 and the solutionsu(l^1 )andu(l^2 )to the
radial Schrödinger equation for these two potentials (at the same energy):
[

^2
2 m

d^2

dr^2

+

(

E−V 1 (r)−


^2 l(l+ 1 )

2 mr^2

)]

u(l^1 )(r)= 0

[


^2

2 m

d^2

dr^2

+

(

E−V 2 (r)−


^2 l(l+ 1 )

2 mr^2

)]

u(l^2 )(r)=0.

(a) Show that by multiplying the first equation from the left byu(l^2 )(r)and the second

one from the left byu(l^1 )(r)and then subtracting, it follows that:

∫L

0

dru(l^2 )(r)[V 1 (r)−V 2 (r)]u(l^1 )(r)=


^2

2 m

[

u(l^2 )(L)

∂u(l^1 )(L)

∂r

−u(l^1 )(L)

∂u(l^2 )(L)

∂r

]

.

(b) IfVi→0 for larger, then both solutions are given for largerby

sin[kr−(lπ/ 2 )+δ(li)]/k. Show that from this it follows that:

∫∞

0

dru(l^2 )(r)[V 1 (r)−V 2 (r)]u(l^1 )(r)=


^2

2 mk

sin(δl(^2 )−δl(^1 )).

Now takeV 1 ≡0 andV 2 ≡Vsmall everywhere. In that case,u(l^1 )andu(l^2 )on the left

hand side can both be approximated byrjl(kr), so that we obtain:

δl≈−

2 mk


^2

∫∞

0

drr^2 j^2 l(kr)V(r).

This is theBorn approximationfor the phase shift. This approximation works well for

potentials that are small with respect to the energy.

(c) [C] Write a (very simple) routine for calculating this integral (or use a library

routine). Of course, it is sufficient to carry out the integration up tormaxsince

beyond that rangeV≡0. Compare the Born approximation with the solution of

the program developed in the previous problem. For the potential, take a weak

Gaussian well:

V(r)=−Aexp[−(r− 1 )^2 ], x<rmax

and

V(r)=0, x≥rmax.

withA=0.01 andrmaxchosen suitably. Result?

(d) Now consider the analysis of items (a) and (b) whereV 1 is the Lennard–Jones

potential without cut-off andV 2 with cut-off. Show that the phase shift for the