2.5 Numerical Applications
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the Fourier transform, the information contained in the coefficients of the expansion
(time-dependent) is represented as an energy-dependent form from which you can
derive the matrix S that can be used to evaluate any experimental information you
want (distributions, cross sections and coefficients of velocity reaction).
2.6 Problems
2.6.1 Qualitative Problems
1. Phase shifts Carefully explain why the phase shifts should approach zero as the
angular momentum parameter goes to ∞.
2. Infinite square well What is the l = 0 phase shift for the infinite square well?
3. Negative phase shifts Explain why the phase shifts are negative for a purely
repulsive potential.
2.6.2 Quantitative Problems
1. Positronium Positronium is similar to the Hydrogen atom except the proton is
replaced with a positron (an antielectron). The positron has the same mass of the
electron but its charge is positive q e instead of negative. When we discussed the
energies of the hydrogen atom, we used the mass of the electron for the reduced
mass of the system. This is a good approximation because the mass of the proton
is much larger than the mass of the electron. For positronium, the masses of
the two particles are identical so one must use the reduced mass. Calculate the
energy levels for the first 5 values of principal quantum number n = 1, 2, 3, 4, 5.
2. H 2 and HD molecules From the vibrational spectrum (see http://webbook.nist.
gov/cgi/cbook.cgi?ID=C1333740&Mask=1000), determine reasonable values
of D e and α for the Morse oscillator. Use the rotational energy spacing to determine r e . You have completely parameterized the Morse potential for H 2 . Now
calculate the Morse energy levels and compare with the experimental data provided. What are the energy levels for HD?
3. Numerov propagator Write a computer program to solve the time-independent
Schrodinger equation. To test your code set the angular momentum and the
potential to zero. The exact solutions are sin kr. Divide the range r = [0, 3π] into
100 equally spaced steps. Use the initial conditions u(0) = 0 and u(h) = sin kh
where h is the equally spaced step size and k =
√
2μE. Compare your exact
solutions with the numerical solutions. Now try several different values of the
angular momentum. Compare your solutions with the Riccatti–Bessel functions
(Download a Bessel function routine from netlib.org). In both cases, the phase
79
the Fourier transform, the information contained in the coefficients of the expansion
(time-dependent) is represented as an energy-dependent form from which you can
derive the matrix S that can be used to evaluate any experimental information you
want (distributions, cross sections and coefficients of velocity reaction).
2.6 Problems
2.6.1 Qualitative Problems
1. Phase shifts Carefully explain why the phase shifts should approach zero as the
angular momentum parameter goes to ∞.
2. Infinite square well What is the l = 0 phase shift for the infinite square well?
3. Negative phase shifts Explain why the phase shifts are negative for a purely
repulsive potential.
2.6.2 Quantitative Problems
1. Positronium Positronium is similar to the Hydrogen atom except the proton is
replaced with a positron (an antielectron). The positron has the same mass of the
electron but its charge is positive q e instead of negative. When we discussed the
energies of the hydrogen atom, we used the mass of the electron for the reduced
mass of the system. This is a good approximation because the mass of the proton
is much larger than the mass of the electron. For positronium, the masses of
the two particles are identical so one must use the reduced mass. Calculate the
energy levels for the first 5 values of principal quantum number n = 1, 2, 3, 4, 5.
2. H 2 and HD molecules From the vibrational spectrum (see http://webbook.nist.
gov/cgi/cbook.cgi?ID=C1333740&Mask=1000), determine reasonable values
of D e and α for the Morse oscillator. Use the rotational energy spacing to determine r e . You have completely parameterized the Morse potential for H 2 . Now
calculate the Morse energy levels and compare with the experimental data provided. What are the energy levels for HD?
3. Numerov propagator Write a computer program to solve the time-independent
Schrodinger equation. To test your code set the angular momentum and the
potential to zero. The exact solutions are sin kr. Divide the range r = [0, 3π] into
100 equally spaced steps. Use the initial conditions u(0) = 0 and u(h) = sin kh
where h is the equally spaced step size and k =
√
2μE. Compare your exact
solutions with the numerical solutions. Now try several different values of the
angular momentum. Compare your solutions with the Riccatti–Bessel functions
(Download a Bessel function routine from netlib.org). In both cases, the phase