1. Show that in the Born approximation, the differentia] cross
section obtained from the negative of a given potential −V(r) is exactly
the same as that obtained from the potential V(r). Get solution
2. An incident particle with mass m, velocity υ, and charge ze scatters off of a charge Ze at the origin. Use the Born approximation to calculate the differential scattering cross section for the screened Coulomb potential...Then let d → ∞, so that V(r) approaches the normal Coulomb potential, and show that dσ/dΩapproaches the Rutherford scattering differential cross section... Get solution
3. (a) A particle with charge +e is incident on an electric dipole consisting of a charge of +e and a charge of −e separated by the vector d (which runs from − e to +e). The energy of the incident particle is sufficiently large to treat the dipole as a small perturbation. Calculate the differential scattering cross section dσ/dΩ as a function of the initial wave vector ki, the scattered wave vector kf, and the standard Rutherford scattering cross section (dσ/dΩ)R, given by...where K = kf −ki and Vc(r) is the Coulomb potential.(b) In the limits kid ≪ 1 and kid ≫ 1, determine whether the dipole differential cross section is larger or smaller than the Rutherford differential cross section. Explain the physical reason for these results. Get solution
4. A particle which is travelling in the +z direction scatters off of a potential consisting of four delta functions at the vertices of a square in the x-y plane at the points (−a, 0, 0), (+a, 0, 0), (0, −a, 0), and (0, +a, 0)....The potential is...where A is a constant. Use the Born approximation to calculate the differential scattering cross section. Express the answer in terms of the magnitude of the incident wave vector k and the scattering angles θ and ϕ. (This is an example where the cross section does depend on ϕ.) Get solution
5. (a) A particle of mass m and energy E scatters off of the central potential V(r) = Ar−2, where A is a constant. Use the Born approximation to calculate the differential cross section dσ/dΩas a function of E and the scattering angle θ.(b) Show that the total cross section σ is infinite. Get solution
6. A particle of mass m is incident on the potential ..., where V0 and r0 are constants with units of energy and length, respectively. The potential is independent of θ and ϕ. The energy of the particle is large, so that the potential can be treated as a small perturbation. Calculate the differential scattering cross section dσ/dΩ. Express the final answer as a function of the scattering angle θ and the energy of the particle E. Get solution
7. (a) A particle with mass m scatters off of the potential...where A is a constant. In other words, this potential forms a square in the x-y plane and is infinitesimally thin in the z direction. Use the Born approximation to calculate the differential scattering cross section. (Assume an arbitrary direction for the incident particle, and express the answer in terms of the momentum transfer vector K = kf − ki.)(b) Show that in the limit where a → ∞ (so that the scattering potential occupies the entire x-y plane), the resulting differential cross section corresponds to only two possible results: either the particle will pass through without any scattering or else it will scatter with the angle of incidence equal to the angle of reflection. Get solution
8. Suppose a particle with mass m scatters off of a finite spherical potential of radius R given by...In the limit where the energy of the incident particle is small, show that the total cross section is...where .... Get solution
2. An incident particle with mass m, velocity υ, and charge ze scatters off of a charge Ze at the origin. Use the Born approximation to calculate the differential scattering cross section for the screened Coulomb potential...Then let d → ∞, so that V(r) approaches the normal Coulomb potential, and show that dσ/dΩapproaches the Rutherford scattering differential cross section... Get solution
3. (a) A particle with charge +e is incident on an electric dipole consisting of a charge of +e and a charge of −e separated by the vector d (which runs from − e to +e). The energy of the incident particle is sufficiently large to treat the dipole as a small perturbation. Calculate the differential scattering cross section dσ/dΩ as a function of the initial wave vector ki, the scattered wave vector kf, and the standard Rutherford scattering cross section (dσ/dΩ)R, given by...where K = kf −ki and Vc(r) is the Coulomb potential.(b) In the limits kid ≪ 1 and kid ≫ 1, determine whether the dipole differential cross section is larger or smaller than the Rutherford differential cross section. Explain the physical reason for these results. Get solution
4. A particle which is travelling in the +z direction scatters off of a potential consisting of four delta functions at the vertices of a square in the x-y plane at the points (−a, 0, 0), (+a, 0, 0), (0, −a, 0), and (0, +a, 0)....The potential is...where A is a constant. Use the Born approximation to calculate the differential scattering cross section. Express the answer in terms of the magnitude of the incident wave vector k and the scattering angles θ and ϕ. (This is an example where the cross section does depend on ϕ.) Get solution
5. (a) A particle of mass m and energy E scatters off of the central potential V(r) = Ar−2, where A is a constant. Use the Born approximation to calculate the differential cross section dσ/dΩas a function of E and the scattering angle θ.(b) Show that the total cross section σ is infinite. Get solution
6. A particle of mass m is incident on the potential ..., where V0 and r0 are constants with units of energy and length, respectively. The potential is independent of θ and ϕ. The energy of the particle is large, so that the potential can be treated as a small perturbation. Calculate the differential scattering cross section dσ/dΩ. Express the final answer as a function of the scattering angle θ and the energy of the particle E. Get solution
7. (a) A particle with mass m scatters off of the potential...where A is a constant. In other words, this potential forms a square in the x-y plane and is infinitesimally thin in the z direction. Use the Born approximation to calculate the differential scattering cross section. (Assume an arbitrary direction for the incident particle, and express the answer in terms of the momentum transfer vector K = kf − ki.)(b) Show that in the limit where a → ∞ (so that the scattering potential occupies the entire x-y plane), the resulting differential cross section corresponds to only two possible results: either the particle will pass through without any scattering or else it will scatter with the angle of incidence equal to the angle of reflection. Get solution
8. Suppose a particle with mass m scatters off of a finite spherical potential of radius R given by...In the limit where the energy of the incident particle is small, show that the total cross section is...where .... Get solution
