Computational Physics - Department of Physics

9.5 Exercises 299

d^2 u(r)

dr^2

+

m

h ̄^2

(V 0 +E)u(r) = 0 r<a, (9.29)

and
d^2 u(r)
dr^2
+m
h ̄^2

E u(r) = 0 r>a. (9.30)

1. We are now going to search for eventual bound states, i.e.,E< 0. The deuteron has only
   one bound state at energyE=− 2. 223 MeV. Discuss the boundary conditions on the wave
   function and use these to show that the solution to the SE is

u(r) =Asin(kr) r<a, (9.31)

and

u(r) =Bexp(−βr) r>a, (9.32)

whereAandBare constants. We have also defined

k=

√

m(V 0 −|E|)/h ̄, (9.33)

and

β=

√

m|E|/h ̄. (9.34)

Show then, using the continuity requirement on the wave function that atr=ayou obtain

the transcendental equation

kcot(ka) =−β. (9.35)

2. Insert values ofV 0 = 60 MeV anda= 1. 45 fm (1 fm = 10−^15 m) and make a plot of Eq. (9.35)
   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. (9.35) is satisfied, obtain a
   numerical value forEusing for example Newton-Raphson’s method or similar methods,
   see chapter 4. To use these functions you need to provide the functionkcot(ka)+βand its
   derivative as function ofE.
   What is smallest possible value ofV 0 which gives one bound state only?


3. Write a program which implements the Green’s function method using Numerov’s method
   for this potential and find the lowest eigenvalue for the casethatV 0 supports only one
   bound state. Use the results from b) to guide your choice of trial eigenvalues. Plot the
   wave function and discuss your results.


4. We turn now to a fitted interaction which reproduces the low-lying phase shifts for scat-
   tering between a proton and neutron. The parametrized version of this potential fits the
   experimental phase-shifts. It is given by

V(r) =Vae

−ax

x

+Vbe

−bx

x

+Vce

−cx

x

(9.36)

withx=μr,μ= 0. 7 fm−^1 (the inverse of the pion mass),Va=− 10. 463 MeV anda= 1 ,

Vb=− 1650. 6 MeV andb= 4 andVc= 6484. 3 MeV andc= 7. Replace the box potential from

point c) and find the wave function and possible eigenvalues for this potential as well.

Discuss your results.