80
2 The Quantum Approach to the Two-Body Problem
shifts should be zero. Finally, use the potential which generated the phase shift
in Fig. 2.7. How do your phases shift compare with those?
4. Differential and integrated cross sections Either use the calculated phase shifts
from the previous problem or the phase shift provided in Fig. 2.7 to calculate the
differential scattering cross section and the integrated cross section.
5. LJ and Morse potentials Using the LJ and Morse potentials you used in Chap. 1
(problem 3) to calculate the phase shifts, differential cross section, and integrated
cross section at a scattering energy equal to D e /2. Can you physically explain the
differences in your results? Considering the long-range behavior (van der Waals)
of the interaction between two atoms predict which potential would produce the
more accurate results for small collision energies.
6. JWKB Phase shifts Use the supplementary FORTRAN code to calculate the
JWKB phase shifts for the potentials in the previous problem. How do they
compare with the Numerov results. Calculate the differential cross section and
the integrated cross section.
7. Homonuclear Diatomic molecules When the two atoms are identical one must
account for this symmetry. If the two atoms are bosons the scattering amplitude
is
f (θ) + f (θ − π)
/2. Likewise if the two atoms are fermions the scattering
amplitude is
f (θ) − f (θ − π)
/2. Use the Morse potential parameters of the
previous two problems and then plot the differential cross sections for bosons,
fermions, and nonidentical atoms.
8. Scattering length and effective range Calculate the s-wave (l = 0) scattering
length and effective range of the two potentials in the previous two problems by
decreasing k and extrapolating to k = 0.
9. Sutherland potential Find the analytical solution to the eigenvalues of the attractive Sutherland potential
10. Harmonic oscillator energy levels You need to be very careful in finding the
eigenvalues and eigenfunctions of any bound state. All numerical propagators
are numerically stable when the wave function is exponentially increasing or
oscillating. However, when the desired eigenfunction is exponentially decreasing you encounter numerical instability. Choose the lowest energy level for a
harmonic oscillator numerically propagate your solution from x = 0 to larger
values of x. You should notice that your numerical solution follows the exact
solution reasonably well until x is greater than the right turning point. Your solution will continue to follow the exact results for steps and then rapidly diverge to
±∞. You need to stop your propagation as soon as it starts to diverge. A stable
way to find the eigenvalues of any bound state is to propagate from r = 0 to
r = r mid and also propagate from r = r max to r = r mid and then make sure the
wave function and the first derivative of the wave function are equal at r = r mid .
In this way, the propagators are always propagating in the numerically stable
direction.
11. Morse oscillator energy levels Follow the hints given in the previous problem
to determine the eigenvalues and eigenfunctions of the Morse oscillator.
12. Distributed Approximating Function—DAF The supplementary material has
FORTRAN subroutines to calculate the DAF approximate for the kinetic energy.
2 The Quantum Approach to the Two-Body Problem
shifts should be zero. Finally, use the potential which generated the phase shift
in Fig. 2.7. How do your phases shift compare with those?
4. Differential and integrated cross sections Either use the calculated phase shifts
from the previous problem or the phase shift provided in Fig. 2.7 to calculate the
differential scattering cross section and the integrated cross section.
5. LJ and Morse potentials Using the LJ and Morse potentials you used in Chap. 1
(problem 3) to calculate the phase shifts, differential cross section, and integrated
cross section at a scattering energy equal to D e /2. Can you physically explain the
differences in your results? Considering the long-range behavior (van der Waals)
of the interaction between two atoms predict which potential would produce the
more accurate results for small collision energies.
6. JWKB Phase shifts Use the supplementary FORTRAN code to calculate the
JWKB phase shifts for the potentials in the previous problem. How do they
compare with the Numerov results. Calculate the differential cross section and
the integrated cross section.
7. Homonuclear Diatomic molecules When the two atoms are identical one must
account for this symmetry. If the two atoms are bosons the scattering amplitude
is
f (θ) + f (θ − π)
/2. Likewise if the two atoms are fermions the scattering
amplitude is
f (θ) − f (θ − π)
/2. Use the Morse potential parameters of the
previous two problems and then plot the differential cross sections for bosons,
fermions, and nonidentical atoms.
8. Scattering length and effective range Calculate the s-wave (l = 0) scattering
length and effective range of the two potentials in the previous two problems by
decreasing k and extrapolating to k = 0.
9. Sutherland potential Find the analytical solution to the eigenvalues of the attractive Sutherland potential
10. Harmonic oscillator energy levels You need to be very careful in finding the
eigenvalues and eigenfunctions of any bound state. All numerical propagators
are numerically stable when the wave function is exponentially increasing or
oscillating. However, when the desired eigenfunction is exponentially decreasing you encounter numerical instability. Choose the lowest energy level for a
harmonic oscillator numerically propagate your solution from x = 0 to larger
values of x. You should notice that your numerical solution follows the exact
solution reasonably well until x is greater than the right turning point. Your solution will continue to follow the exact results for steps and then rapidly diverge to
±∞. You need to stop your propagation as soon as it starts to diverge. A stable
way to find the eigenvalues of any bound state is to propagate from r = 0 to
r = r mid and also propagate from r = r max to r = r mid and then make sure the
wave function and the first derivative of the wave function are equal at r = r mid .
In this way, the propagators are always propagating in the numerically stable
direction.
11. Morse oscillator energy levels Follow the hints given in the previous problem
to determine the eigenvalues and eigenfunctions of the Morse oscillator.
12. Distributed Approximating Function—DAF The supplementary material has
FORTRAN subroutines to calculate the DAF approximate for the kinetic energy.