Computational Physics - Department of Physics

96 4 Non-linear Equations

and

u(r) =Bexp(−

√

2 m|E|r/h ̄) r>a, (4.6)

whereAandBare constants. Using the continuity requirement on the wavefunction atr=a

one obtains the transcendental equation

√

2 m(V 0 −|E|)cot(

√

2 ma^2 (V 0 −|E|)/ ̄h) =−

√

2 m|E|. (4.7)

This equation is an example of the kind of equations which could be solved by some of the

methods discussed below. The algorithms we discuss are the bisection method, the secant

and Newton-Raphson’s method.

In order to find the solution for Eq. (4.7), a simple procedureis to define a function

f(E) =

√

2 m(V 0 −|E|)cot(

√

2 ma^2 (V 0 −|E|)/ ̄h)+

√

2 m|E|. (4.8)

and with chosen or given values foraandV 0 make a plot of this function and find the ap-

proximate region along theE−axiswheref(E) = 0. We show this in Fig. 4.1 forV 0 = 20 MeV,

a= 2 fm andm= 938 MeV. Fig. 4.1 tells us that the solution is close to|E|≈ 2. 2 (the binding

-100

-50

0

50

100

0 1 2 3 4 5

f(E)[MeV]

|E|[MeV]

f(x)

Fig. 4.1Plot off(E)in Eq. (4.8) as function of energy |E| in MeV. Te functionf(E)is in units of megaelectron-

volts MeV. Note well that the energyEis for bound states.

energy of the deuteron). The methods we discuss below are then meant to give us a numer-

ical solution forEwhere f(E) = 0 is satisfied and withEdetermined by a given numerical

precision.

4.2 Iterative Methods

To solve an equation of the typef(x) = 0 means mathematically to find all numberss^1 so that

f(s) = 0. In all actual calculations we are always limited by a given precision when doing

(^1) In the following discussion, the variablesis reserved for the value ofxwhere we have a solution.