9.2 Shooting methods 289
Table 9.2Integrated and exact solution of the differential equation−y′′+x^2 y= 2 εywith boundary conditions
y(−∞) = 0 andy(∞) = 0.
xi=ihexp(−x^2 / 2 ) y(xi)
-.100000E+02 0.192875E-21 0.192875E-21
-.800000E+01 0.126642E-13 0.137620E-13
-.600000E+01 0.152300E-07 0.157352E-07
-.400000E+01 0.335462E-03 0.331824E-03
-.200000E+01 0.135335E+00 0.128549E+00
0.000000E-00 0.100000E+01 0.912665E+00
0.200000E+01 0.135335E+00 0.118573E+00
0.400000E+01 0.335463E-03 -.165045E-01
0.600000E+01 0.152300E-07 -.250865E+03
0.800000E+01 0.126642E-13 -.231385E+09
0.900000E+01 0.257677E-17 -.101904E+13
y(x) =(
1
π) 1 / 4
exp(−x^2 / 2 ).The reader should observe that this solution is imposed by the boundary conditions, which
again follow from the quantum mechanical properties we require for the solution. We repeat
the integration exercise which we did for the previous example, starting from a large negative
number (x 0 =− 10 , which gives a value for the eigenfunction close to zero) andchoose the
lowest energy and its corresponding eigenfunction. We obtain fory 2
y 2 ≈−y 0 +y 1(
2 +h^2 x^2 −h^2)
,
and using the exact eigenfunction we can replacey 1 with the derivative atx 0. We use now
N= 1000 and integrate our equation fromx 0 =− 10 toxN= 10. The results are shown in Table
9.2 for selected values ofxi. In the beginning of our integrational interval, we obtain an
integrated quantity which resembles the analytic solution, but then our integrated solution
simply explodes and diverges. What is happening? We startedwith the exact solution for both
the eigenvalue and the eigenfunction!
The problem is due to the fact that our differential equationhas two possible solution for
eigenvalues which are very close (− 1 / 2 and+ 1 / 2 ), either
y(x)∼exp(−x^2 / 2 ),or
y(x)∼exp(x^2 / 2 ).
The boundary conditions, imposed by our physics requirements, rule out the last possibility.
However, our algorithm, which is nothing but an approximation to the differential equation
we have chosen, picks up democratically both solutions. Thus, although we start with the
correct solution, when integrating we pick up the undesiredsolution. In the next subsections
we discuss how to cure this problem.
9.2.3 Schrödinger equation for spherical potentials
We discuss the numerical solution of the Schrödinger equation for the case of a particle with
massmmoving in a spherical symmetric potential.
The initial eigenvalue equation reads