4.6 Exercises 107
u(r) =Asin(kr) r<a,and
u(r) =Bexp(−βr) r>a,
whereAandBare constants. We have also defined
k=√
m(V 0 −|E|)/h ̄,and
β=
√
m|E|/h ̄.Show then, using the continuity requirement on the wave function that atr=ayou obtain the
transcendental equation
kcot(ka) =−β. (4.41)
Insert values ofV 0 = 60 MeV anda= 1. 45 fm (1 fm = 10−^15 m) and make a plot plotting
programs) of Eq. (4.41) as function of energyEin order to find eventual eigenvalues. See if
these values result in a bound state forE.
When you have localized on your plot the point(s) where Eq. (4.41) is satisfied, obtain a
numerical value forEusing the class you programmed in the previous exercise, including the
Newton-Raphson’s method, the bisection method and the secant method. Make an analysis
of these three methods and discuss how many iterations are needed to find a stable solution.
What is smallest possible value ofV 0 which gives a bound state?