1. The Lippmann-Schwinger formalism can also be applied to a
one-dimensional transmission-reflection problem with a finite-range
potential, V(x) ≠ 0 for 0 x| a only.a. Suppose we have an incident wave
coming from the left: ...... How must we handle the singular 1/(E − H0)
operator if we are to have a transmitted wave only for x > a and a
reflected wave and the original wave for x ? Is the E → E + iε
prescription still correct? Obtain an expression for the appropriate
Green’s function and write an integral equation for ⟨x|ψ(+)⟩.b. Consider
the special case of an attractive δ-function potential...Solve the
integral equation to obtain the transmission and reflection amplitudes.
Check your results with Gottfried 1966, 52.c. The one-dimensional
δ-function potential with γ > 0 admits one (and only one) bound state
for any value of γ. Show that the transmission and reflection
amplitudes you computed have bound-state poles at the expected positions
when k is regarded as a complex variable. Get solution
2. Prove...in each of the following ways.a. By integrating the differential cross section computed using the first-order Born approximation.b. By applying the optical theorem to the forward-scattering amplitude in the second-oider Born approximation. [Note that f(0) is real if the first-order Born approximation is used.] Get solution
3. Consider a potential...where V0 may be positive or negative. Using the method of partial waves, show that for |V0| ≪ E = ℏ2k2/2m and kR ≪ 1 the differential cross section is isotropic and that the total cross section is given by...Suppose the energy is raised slightly. Show that the angular distribution can then be written as...Obtain an approximate expression for B/A. Get solution
4. A spinless particle is scattered by a weak Yukawa potential...where µ > 0 but V0 can be positive or negative. It was shown in the text that the first-order Born amplitude is given by...a. Using f(1)(θ) and assuming |δl| ≪ 1, obtain an expression for δl in terms of a Legendre function of the second kind,...b. Use the expansion formula...to prove each assertion.(i) δl is negative (positive) when the potential is repulsive (attractive).(ii) When the de Broglie wavelength is much longer than the range of the potential, δl is proportional to k2l + 1. Find the proportionality constant. Get solution
5. Check explicitly the x − px uncertainty relation for the ground state of a particle confined inside a hard sphere: V = ∞ for r > a, V = 0 for r (Hint: Take advantage of spherical symmetry.) Get solution
6. Consider the scattering of a particle by an impenetrable sphere...a. Derive an expression for the s-wave (l = 0) phase shift. (You need not know the detailed properties of the spherical Bessel functions to be able to do this simple problem!)b. What is the total cross section σ[σ = ∫(dσ/dΩ)dΩ] in the extreme low-energy limit k → 0? Compare your answer with the geometric cross section πa2. You may assume without proof:... Get solution
7. Use δl =Δ(b)|b = l/k to obtain the phase shift δl for scattering at high energies by (a) the Gaussian potential, V = V0exp( − r2/a2), and (b) the Yukawa potential, V = V0exp(−µr)/µr. Verify the assertion that δl goes to zero very rapidly with increasing l (k fixed) for l ≫ kR, where R is the “range” of the potential. [The formula for Δ(b) is given in (7.4.14)]. Get solution
8. a. Prove...where r (r>) stands for the smaller (larger) of r and r'.b. For spherically symmetric potentials, the Lippmann-Schwinger equation can be written for spherical waves:...Using (a), show that this equation, written in the x-representation, leads to an equation for the radial function, Al(k; r), as follows:...By taking r very large, also obtain... Get solution
9. Consider scattering by a repulsive δ-shell potential:...a. Set up an equation that determines the s-wave phase shift δ0 as a function of k (E = ℏk2/2m).b. Assume now that γ is very large,...Show that if tan kR is not close to zero, the s-wave phase shift resembles the hard-sphere result discussed in the text. Show also that for tan kR close to (but not exactly equal to) zero, resonance behavior is possible; that is, cot δ0 goes through zero from the positive side as k increases. Determine approximately the positions of the resonances keeping terms of order 1/γ; compare them with the bound-state energies for a particle confined inside a spherical wall of the same radius,...Also obtain an approximate expression for the resonance width Γdenned by...and notice, in particular, that the resonances become extremely sharp as γ becomes large, (Note: For a different, more sophisticated approach to this problem see Gottfried 1966, 131−141, who discusses, the analytic properties of the Dl-function defined by Al = jl/Dl.) Get solution
10. A spinless particle is scattered by a time-dependent potential...Show that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or decreased by ℏω. Obtain dσ/dΩ. Discuss qualitatively what happens if the higher-order terms are taken into account. Get solution
11. Show that the differential cross section for the elastic scattering of a fast electron by the ground state of the hydrogen atom is given by...(Ignore the effect of identity.) Get solution
12. Let the energy of a particle moving in a central field be E(J1J2J3), where (J1, J2, J3) are the three action variables. How does the functional form of E specialize for the Coulomb potential? Using the recipe of the action-angle method, compare the degeneracy of the central field and the Coulomb problems and relate it to the vector A.If the Hamiltonian is...how are these statements changed?Describe the corresponding degeneracies of the central field and Coulomb problems in quantum theory in terms of the usual quantum numbers (n, l, m) and also in terms of the quantum numbers (k, m, n). Here the second set, (k, m, n), labels the wave functions ...How are the wave functions ... related to Laguerre times spherical harmonics? Get solution
2. Prove...in each of the following ways.a. By integrating the differential cross section computed using the first-order Born approximation.b. By applying the optical theorem to the forward-scattering amplitude in the second-oider Born approximation. [Note that f(0) is real if the first-order Born approximation is used.] Get solution
3. Consider a potential...where V0 may be positive or negative. Using the method of partial waves, show that for |V0| ≪ E = ℏ2k2/2m and kR ≪ 1 the differential cross section is isotropic and that the total cross section is given by...Suppose the energy is raised slightly. Show that the angular distribution can then be written as...Obtain an approximate expression for B/A. Get solution
4. A spinless particle is scattered by a weak Yukawa potential...where µ > 0 but V0 can be positive or negative. It was shown in the text that the first-order Born amplitude is given by...a. Using f(1)(θ) and assuming |δl| ≪ 1, obtain an expression for δl in terms of a Legendre function of the second kind,...b. Use the expansion formula...to prove each assertion.(i) δl is negative (positive) when the potential is repulsive (attractive).(ii) When the de Broglie wavelength is much longer than the range of the potential, δl is proportional to k2l + 1. Find the proportionality constant. Get solution
5. Check explicitly the x − px uncertainty relation for the ground state of a particle confined inside a hard sphere: V = ∞ for r > a, V = 0 for r (Hint: Take advantage of spherical symmetry.) Get solution
6. Consider the scattering of a particle by an impenetrable sphere...a. Derive an expression for the s-wave (l = 0) phase shift. (You need not know the detailed properties of the spherical Bessel functions to be able to do this simple problem!)b. What is the total cross section σ[σ = ∫(dσ/dΩ)dΩ] in the extreme low-energy limit k → 0? Compare your answer with the geometric cross section πa2. You may assume without proof:... Get solution
7. Use δl =Δ(b)|b = l/k to obtain the phase shift δl for scattering at high energies by (a) the Gaussian potential, V = V0exp( − r2/a2), and (b) the Yukawa potential, V = V0exp(−µr)/µr. Verify the assertion that δl goes to zero very rapidly with increasing l (k fixed) for l ≫ kR, where R is the “range” of the potential. [The formula for Δ(b) is given in (7.4.14)]. Get solution
8. a. Prove...where r (r>) stands for the smaller (larger) of r and r'.b. For spherically symmetric potentials, the Lippmann-Schwinger equation can be written for spherical waves:...Using (a), show that this equation, written in the x-representation, leads to an equation for the radial function, Al(k; r), as follows:...By taking r very large, also obtain... Get solution
9. Consider scattering by a repulsive δ-shell potential:...a. Set up an equation that determines the s-wave phase shift δ0 as a function of k (E = ℏk2/2m).b. Assume now that γ is very large,...Show that if tan kR is not close to zero, the s-wave phase shift resembles the hard-sphere result discussed in the text. Show also that for tan kR close to (but not exactly equal to) zero, resonance behavior is possible; that is, cot δ0 goes through zero from the positive side as k increases. Determine approximately the positions of the resonances keeping terms of order 1/γ; compare them with the bound-state energies for a particle confined inside a spherical wall of the same radius,...Also obtain an approximate expression for the resonance width Γdenned by...and notice, in particular, that the resonances become extremely sharp as γ becomes large, (Note: For a different, more sophisticated approach to this problem see Gottfried 1966, 131−141, who discusses, the analytic properties of the Dl-function defined by Al = jl/Dl.) Get solution
10. A spinless particle is scattered by a time-dependent potential...Show that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or decreased by ℏω. Obtain dσ/dΩ. Discuss qualitatively what happens if the higher-order terms are taken into account. Get solution
11. Show that the differential cross section for the elastic scattering of a fast electron by the ground state of the hydrogen atom is given by...(Ignore the effect of identity.) Get solution
12. Let the energy of a particle moving in a central field be E(J1J2J3), where (J1, J2, J3) are the three action variables. How does the functional form of E specialize for the Coulomb potential? Using the recipe of the action-angle method, compare the degeneracy of the central field and the Coulomb problems and relate it to the vector A.If the Hamiltonian is...how are these statements changed?Describe the corresponding degeneracies of the central field and Coulomb problems in quantum theory in terms of the usual quantum numbers (n, l, m) and also in terms of the quantum numbers (k, m, n). Here the second set, (k, m, n), labels the wave functions ...How are the wave functions ... related to Laguerre times spherical harmonics? Get solution
