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# Exercises

1. Justify Equation (10.5).

2. Justify Equation (10.7).

3. Consider the Green's function for the Helmholtz equation:

1. Assuming, as seems reasonable, that , where , demonstrate that the two independent solutions in the region are

2. Show that

where is a sphere of vanishing radius centered on . Hence, deduce that is the required Green's function.

1. Demonstrate that

where .

2. Show that

3. Finally, by taking the limit , conclude that

where is a Dirac delta function.

4. Demonstrate that the probability current associated with the spherical waves

where , is

Hence, deduce that and correspond to outgoing and ingoing spherical waves, respectively.

5. Justify Equation (10.37).

6. Show that, in the Born approximation, the total scattering cross-section associated with the Yukawa potential, (10.35), is

7. Consider a scattering potential of the form

Demonstrate, using the Born approximation, that

and

8. Show that the differential cross-section for the elastic scattering of a fast electron by the ground state of a hydrogen atom is

where , and is the Bohr radius.

9. Justify Equations (10.57) and (10.59).

10. Demonstrate that, for a spherically symmetric scattering potential,

in each of the following ways:
1. By directly integrating the differential scattering cross-section obtained from the Born approximation.
2. By utilizing the optical theorem in combination with the first two terms in the Born expansion. [Note that the first term in this expansion is real, and, therefore, does not contribute to .]

11. Justify Equations (10.98) and (10.100).

12. Consider a scattering potential that takes the constant value for , and is zero for , where may be either positive or negative. Using the method of partial waves, show that for , and ,

and

13. Consider scattering of particles of mass and incident wavenumber by a repulsive -shell potential:

where . Show that the S-wave phase-shift is given by

Assuming that , demonstrate that if then the solution of the previous equation takes the form

Of course, in the limit , the preceding equation yields , which is the same result obtained when particles are scattered by a hard sphere of radius . (See Section 10.8.) This is not surprising, because a strong repulsive -shell potential is indistinguishable from hard sphere as far as external particles are concerned.

The previous solution breaks down when , where is a positive integer. Suppose that

where . Demonstrate that the S-wave contribution to the total scattering cross-section takes the form

where

Hence, deduce that the net S-wave contribution to the total scattering cross-section is

Obviously, there are resonant contributions to the cross-section whenever . Note that the are the possible energies of particles trapped within the -shell potential. Hence, the resonances are clearly associated with incident particles tunneling though the -shell and forming transient trapped states. However, the width of the resonances (in energy) decreases strongly as the strength, , of the shell increases.

14. Consider the mutual scattering of two, counter-propagating, unpolarized, proton beams in the center of mass frame. Making use of the Born approximation, demonstrate that the differential scattering cross-section is

where is the scattering angle, the proton mass, and the magnitude of the incident momenta of protons in each beam.

Next: Relativistic Electron Theory Up: Scattering Theory Previous: Scattering of Identical Particles
Richard Fitzpatrick 2016-01-22