Computational Physics

602 Appendix A

We shall take
^2 / 2 m=1 in the following. This equation can be solved analytically

(it can be mapped onto the Laplace equation which plays an important role in the

solution of the hydrogen atom [25]), the bound state energies are given by

En=−V 0

(

1 −

n+^12

√

V 0

)

, n=0, 1, 2,...

In this problem it is assumed that you have a routine available for integrating the

one-dimensional Schrödinger equation: you can write a Numerov routine or use a

library routine. We want to determine the bound state spectrum numerically. This is

done in the following way. First, a rangexmaxis defined, beyond which we can safely

approximate the potential byV(x)=0;xmax≈10 is a good value for this range. For

some energyE, the Numerov routine can be used to integrate the Schrödinger

equation numerically up to the rangexmaxyielding a solutionu(x), and beyondxmax,

the solution, which we denote asv(x), is known analytically:

v(x)=Ae−qx+Beqx,

q=

√

−E.

Consider theWronskian W:

W=u′(xmax)v(xmax)−u(xmax)v′(xmax)

(the prime′denotes a derivative). For a bound state, the coefficientBin the solutionv

should vanish and we have for the Wronskian:

WB= 0 =[u′(xmax)+qu(xmax)]e−qxmax.

The matching condition between the analytical solutionvand the numerical one (u),

is equivalent to the Wronskian becoming equal to zero, and we thus have to find the

energies for which the Wronskian withB=0 vanishes. These energies can be found

using, for example, the secant method.

(a) [C] Write a function which uses the Numerov procedure for solving the

Schrödinger equation and returns the Wronskian withB=0 as a function of the

energyE.

(b) [C] Write a code for the secant method ofAppendix A3. Test this with some

simple function you know the roots of, for example sinx.

(c) [C] Use the secant code for finding the energies for which the Wronskian with

B=0 vanishes. Note that the energies lie between 0 and−V 0. The search starts

fromE=−V 0 , and after increasing the energy by a small step (which is

predefined in the program or by the user), it is checked if the Wronskian changes

sign. If this is the case, the secant method is executed for the last energy interval.

When the root is found, the energy is increased again until the Wronskian changes

sign. The procedure is repeated until the energy becomes positive.

Check if the energies match the exact values given above.