4.4 Numerical solution of the Schr ̈odinger equation 69
Fig. 4.6The potential in the central-
field approximation including the term
that is proportional tol(l+1)/r^2 is
drawn here forl=2andthesame
approximate electrostatic VCF(r)as
shown in Fig. 4.4. The functionP(r)=
rR(r) was drawn forn=6andl=2
using the method described in Exer-
cise 4.10.make the assumed potential correspond closely to the real potential, as
described in the next section. Equation 4.12 is a second-order differen-
tial equation and we can numerically calculateP(r), the value of the
function atr,fromtwonearbyvalues,e.g.u(r−δr)andu(r− 2 δr).^1313 The step sizeδrmust be small com-
pared to the distance over which the
wavefunction varies; but the number of
steps must not be so large that round-
off errors begin to dominate.
Thus, working from nearr= 0, the method gives the numerical value of
the function at all points going out as far as is necessary. The region of
the calculation needs to extend beyond the classical turning point(s) by
an amount that depends on the energy of the wavefunction being calcu-
lated. These general features are clearly seen in the plots produced in
Exercise 4.10. Actually, that exercise describes a method of finding the
radial wavefunctionR(r)ratherthanP(r)=rR(r) but similar princi-
ples apply.^14 If you carry out the exercise you will find that the behaviour^14 In a numerical method there is no
reason why we should not calculate the
wavefunction directly;P(r) was intro-
duced to make the equations neater in
the analytical approach.
at largerdepends very sensitively on the energyE—the wavefunction
diverges ifEis not an eigenenergy of the potential—this gives a way
of searching for those eigenenergies. If the wavefunction diverges up-
wards forE′and downwards forE′′then we know that an eigenenergy
of the systemEklies between these two values,E′<Ek<E′′.Test-
ing further values between these upper and lower bounds narrows the
range and gives a more precise value ofEk(as in the Newton–Raphson
method for finding roots). This so-called ‘shooting’ method is the least
sophisticated method of computing wavefunctions and energies, but it
is adequate for illustrating the principles of such calculations. Results
are not given here since they can readily be calculated—the reader is
strongly encouraged to implement the numerical method of solution, us-
ing a spreadsheet program, as described in Exercise 4.10. This shows
how to find the wavefunctions for an electron in an arbitrary potential
and verifies that the energy levels obey a quantum defect formula such
as eqn 4.1 in any potential that is proportional to 1/rat long range (see
Fig. 4.7).