¡Descarga lista de ejercicios QM2 y más Ejercicios en PDF de Mecánica Cuántica solo en Docsity!
40 Problems
- Express, by means of scattering phase shifts, the first three coef- ficients of the expansion of the elastic scattering cross section ;~ in ter~s of Legendre polynomials. @Calculate the differential cross section for scattering in a repul-
sive field U = ~ in the .Born approximation. Jli'Lpcat (he cakalatioIi fQ~
tl:te \ eese sf 61a8sical H'Ieel:taRi@4. Find the limits of applicability of the formulae obtained.
- Find the discrete levels for a particle in the attractive field U(r) = -Uo exp (-r/a) for 1 = O. Find the scattering phase shift 00 for this potential and discuss the relation between 00 and the discrete spectrurn.
- Show that for a Coulomb field there is a one-to-one correspondence between the poles. of the scattering amplitudes and the levels of the discrete spectrum. Hint. Use the relation
e 216 ¡ = r (z +1+{-)
. r(Z+l- !).
t;)Determine how the scattering phase shift changes for a small change in't?e scattering potential. Find the expression for the scattering phase shift in the case in which the potential can be considered as a perturbation. @Calculate the scattering phase shifts of slow particles in the field
V = a/r3. The particles are slow enough for the condition flakJñ^2 -e 1 to
be satisfied.
- Find the total cross section for the elastic scattering of fast par- ticles by a perfectly rigid sphere of radius a (l ~ a, where l is the de Broglie wavelength).
@Find in the Born approximation the differential ánd total cross
sections for scattering in the fields:
e^ -a,
(a) U(r) = g2_
r (b) U(r) = U¿ «<",
(c) U(r) = U¿ e-a,.
11. Using the Born approximation, find the differential and total
cross sections for the elastic scattering of fast electrons (a) by a hydrogen atom, (b) by a heliurn atom.
SEc.20] CONTINUOUS^ EIGENV^ ALUES:^ COLLISION^ THEORY^^121
It is interesting to note that forthe Coulomb field the classical Iimit
is approached for small v, whereas forpotentials that have a finite range
. a, such as are discussed in Seco19, the classicaIlimit ÍB approached when
ea/X)!» 1, that is, for Iarge v. This is because the "size" /ZZ'e^2 /¡Lv^2 / of the Coulomb field increases more rapidly than X = n/ ¡LV as v decreases. Problems
- Show that the coefficients of scattering by a one-dimensional square well poten-
tia! (like Fig. 14 except that Vo < O) are given by Eqs. (17.5) if the sign oí Vo is
changed there and in the expression for 0:. Discuss the dependence of transmission .coefficient on E in this case.
- Show that Eqs. (18.4) and (18.7) are valid for a general binary collision if l' is given by' (18.5); make use of conservation oí energy and mass. , 3. Show that, when a particle of mass mI collides elastically with a particle oí mass m, that is initially at rest, all the recoil (mass m,) particles are scattered in the forward hemisphere in the laboratory coordinate system. If the angular distribution
is spherically symmetrical in the center-of-mass system, what is it for m, in the
laboratory system?
- Express the scattering wave íunction (19.1) outside the scattering potential (but not necessarily in the asymptotic region) as the sum of aplane wave and an infinite series of spherical Hankel functions of the first kind [see Eqs. (15.12)]. From this expression-and the discussion of Eqs. (15.13), show that the scattered wave is purely outgoing, even inside of the asymptotic region. · i.?- , 5.' What must Voa' be for a three-dimensional square well potential in order that the scattering cross section be zero at zero bombarding energy (Ramsauer-Townsend effect)? Find the leading term in the expression for the total cross section for small
bombarding energy. Note that both the l = Oand the l = 1 partial waves must be
inoluded..
- 8tate clearly the assumptíons that go into the derivation of Eq. (19.31), and verify that it is a suitable approximation for the total cross section at low bombarding energies when the l = Owave is in resonance.
- Make use of Eq. (19.31) and the result of Probo 5, Chap. IV, to obtain an approximate expression for the total scattering cross section by a particular potential
ín terms of the bombarding energy E and the binding energy • oí a particle in that
potential, when E and e are small in comparison with Vo. · .;,.. , 8. Compute and make a polar plot oí the dífferential scattering cross section for a perfectly rigid sphere when ka = i, using the first three partial waves (1 = O, 1, 2). What is the total cross section in this case, and what is the approximate accuracy of this~ult when the three terms are used?. ·$. !!;IFind a general expression for the phase shift produced by a scattering potential Ver) = A/r', where A > o. Is the total cross section finite? If not, does the diver= gence come from small or large scattering angles, and why? What modifications are
necessary in the calculation if A < O? Are any difficulties encountered in this latter
case? , 10. Protons of 200,000 electron-volts energy are scattered from alurninum. The directly back scattered intensity (/1 = 180°) is found to be 96 per cent of that com- puted from the Rutherford formula. Assume this to be due to a modification of the Coulomb potential that is of sufficiently short range so that only the phase shift for 1 = O is affected. Is this modification attractive or repulsive? Find the sign and magnitude oí the change in the nhase shift for 1 = Oproduced by the modificatiog,
468 Ch. 16: Scattering
knows only a finite number of phase shifts over a limited range of energy, and this does not allow one to apply either of the theorems.
Further reading for Chapter 16
Goldberger and Watson (1964) has long been regarded as the authoritative
reference on scattering theory, although it has now been superseded to some
extent by Newton (1982). Both of them are research tomes. The beginning
student may find the treatment by Rodberg and Thaler (1967) to be more
accessible. Wu and Ohmura (1962) is intermediate between the textbook and
research levels.
Problems
16.1 Derive Eq. (16.5) from momentum and energy conservation.
@ Calculate the R. = O phase shift for the repulsive 6 shell potential,
V (r) = e 6 (r - a). Determine the conditions under which it will be
approximately equal to the phase shift of a hard sphere of the same radius a, and note the conditions under which it may significantly differ from the hard sphere phase shift even though e is very large. 16.3 Show that \lr(+) and \lr(-), defined by (16.69), have the correct asymp- totic forms, (16.42) and (16.46), respectively. 16.4 Use the Born approximation to calculate the differential cross section
for scattering by the Yukawa potential, V(r) = Va e-ar^ [ccr,
16.5 In Example 1 of Seco 16.4 (spin-spin interaction),assume that the two
particles are an electron and a proton, and add to Ha the magnetic
dipole interaction -B· (J.l~ + J.lp). Calculate the scattering cross sec-
tions in the Born Approximation, taking into account the fact that kinetic energy will not be conserved. 16.6 For Example 1 of Seco 16.4 (without a magnetic field) , assume that the phase shifts for the central potentiaLl::ú(r) are-k-nown, and use the DWBA to calculate the additional scattering due to the spin-spin
interaction Vs (r )0-(1).0-(2). Does skew scattering occur?
16.7 Show that Example 2 of Seco16.4 (spin-orbit interaction) can be solved "exactly" by introducing the total angular momentum eigenfunctions (7.104) as basis functions, and computing a new set of phase shifts that depend upon both the orbital angular mornenturn é and the total angular momentum j. The solution will be as "exact" as the com- putation of the phase shifts. [Ref.: Goldberger and Watson (1964), Seco 7.2.]
