1. Rutherford scattering. An incident particle of charge q1 and
kinetic energy E scatters off a heavy stationary particle of charge
q2.(a) Derive the formula relating the impact parameter to the
scattering angle. Answer: b = (q1q2/8π∑0E) cot(θ/2).(a) Determine the
differential scattering cross-section. Answer:...(c) Show that the total
cross-section for Rutherford scattering is infinite. We say that the
1/r potential has "infinite range"; you can't escape from a Coulomb
force. Get solution
2. Construct the analogs to Equation 11.12 for one-dimensional and two-dimensional scattering. Get solution
3. Prove Equation 11.33, starting with Equation 11.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of l must separately vanish. Get solution
4. Consider the case of low-energyscattering from a spherical delta function shell:V(r) = αδ(r-a)Where α and a are constants. Calculate the scattering amplitude, f(θ), the differenential cross-section, d(θ), and the total cross-section, σ. Assume ka « 1, so that only the l = 0 term contributes significantly. (To simplify matters, throw out all I ≠ 0 terms right from the start.) The main problem, of course, is to determine ao. Express your answer in terms of the dimensionless quantity ß ≡ 2maa/h2. Answer: σ = 4πa2ß2/(l + ß). Get solution
5. A particle of mass m and energy E is incident from the left on the potential...(a) If the incoming wave is Ae'kx (where ...), find the reflected wave.Answer..., where ....(b) Confirm that the reflected wave has the same amplitude as the incident wave.(b) Find the phase shift δ (Equation 1) for a very deep well (E « v0)Answer: δ = —ka.Equation 1Ψo(x) = A(eikx – e-ikx) (V(x) = 0). Get solution
6. What are the partial wave phase shifts ... for hard-sphere scattering (Example 11.3)? ... ... That's the exact answer, but it's not terribly illuminating, so let's consider the limiting case of low-energy scattering: ..., this amounts to saying that the wavelength is much greater than the radius of the sphere.) Referring to Table 4.4, we note that ..., for small z, so ... Get solution
7. Find the S-wave (l = 0) partial wave phase shift δ0(k) for scattering from a delta-function shell (Problem 1). Assume that the ratial wave function u(r) goes to o as r →∞ Answer:...Problem 1Consider the case of low-energyscattering from a spherical delta function shell:V(r) = αδ(r-a)Where α and a are constants. Calculate the scattering amplitude, f(θ), the differenential cross-section, d(θ), and the total cross-section, σ. Assume ka « 1, so that only the l = 0 term contributes significantly. (To simplify matters, throw out all I ≠ 0 terms right from the start.) The main problem, of course, is to determine ao. Express your answer in terms of the dimensionless quantity ß ≡ 2maa/h2. Answer: σ = 4πa2ß2/(l + ß). Get solution
8. Check that Equation 11.65 satisfies Equation 11.52, by direct substitution. Hint: ... ... Get solution
9. Show that the ground state of hydrogen (Equation 4.80) satisfies the integral form of the Schrodinger equation, for the appropriate V and E (note that E is negative, so .... ... Get solution
10. Find the scattering amplitude, in the Born approximation, for soft-sphere scattering at arbitrary energy. Show that your formula reduces to Equation 11.82 in the low-energy limit. ... Get solution
11. Evaluate the integral in Equation 11.91, to confirm the expression on the right. ... Get solution
12. Calculate the total cross-section for scattering from a Yukawa potential, in the Born approximation. Express your answer as a function of E Get solution
13. For the potential in Problem 1,(a) calculate f(θ), D(θ), and σ, in the low-energy Born approximation;(b) calculate f (θ) for arbitrary energies, in the Born approximation;(c) show that your results are consistent with the answer to Problem 11.4, in the appropriate regime.Problem 1Calculate θ (as a function of the impact parameter) for Rutherford scattering, in the impulse approximation. Show that your result is consistent with the exact expression (Problem 1), in the appropriate limit.Problem 1Rutherford scattering. An incident particle of charge q1 and kinetic energy E scatters off a heavy stationary particle of charge q2.(a) Derive the formula relating the impact parameter to the scattering angle. Answer: b = (q1q2/8π∑0E) cot(θ/2). Get solution
14. Calculate θ (as a function of the impact parameter) for Rutherford scattering, in the impulse approximation. Show that your result is consistent with the exact expression (Problem 1), in the appropriate limit.Problem 1Rutherford scattering. An incident particle of charge q1 and kinetic energy E scatters off a heavy stationary particle of charge q2.(a) Derive the formula relating the impact parameter to the scattering angle. Answer: b = (q1q2/8π∑0E) cot(θ/2). Get solution
15. Find the scattering amplitude for low-energy soft-sphere scattering in the second Born approximation. Answer: —(2m Voa3/3h2)[1 - (4mVoa2/5h2)]. Get solution
16. Find the Green's function for the one-dimensional Schrodinger equation, and use it to construct the integral form (analogous to Equation 1). Answer:...Equation 1... Get solution
17. Use your result in problem 1 to develop the Born approximation for one-dimensional scattering (on the interval -∞ ...problem 1Find the Green's function for the one-dimensional Schrodinger equation, and use it to construct the integral form (analogous to Equation 1). Answer:...Equation 1... Get solution
18. Use the one-dimensional Born approximation (Problem 11.17) to compute the transmission coefficient (T = 1 — R) for scattering from a delta function (Equation 2.114) and from a finite square well (Equation 2.145). Compare your results with the exact answers (Equations 2.141 and 2.169). V (x) = -αδ(x) Get solution
19. Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude....Hint : Use Equations 1 and 2.Equation 1...Equation 2... Get solution
20. Use the Bom approximation to determine the total cross-section for scattering from a gaussian potential...Express your answer in terms of the constants A, μ, and m (the mass of the incident particle), and k ≡ ..., where E is the incident energy. Get solution
2. Construct the analogs to Equation 11.12 for one-dimensional and two-dimensional scattering. Get solution
3. Prove Equation 11.33, starting with Equation 11.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of l must separately vanish. Get solution
4. Consider the case of low-energyscattering from a spherical delta function shell:V(r) = αδ(r-a)Where α and a are constants. Calculate the scattering amplitude, f(θ), the differenential cross-section, d(θ), and the total cross-section, σ. Assume ka « 1, so that only the l = 0 term contributes significantly. (To simplify matters, throw out all I ≠ 0 terms right from the start.) The main problem, of course, is to determine ao. Express your answer in terms of the dimensionless quantity ß ≡ 2maa/h2. Answer: σ = 4πa2ß2/(l + ß). Get solution
5. A particle of mass m and energy E is incident from the left on the potential...(a) If the incoming wave is Ae'kx (where ...), find the reflected wave.Answer..., where ....(b) Confirm that the reflected wave has the same amplitude as the incident wave.(b) Find the phase shift δ (Equation 1) for a very deep well (E « v0)Answer: δ = —ka.Equation 1Ψo(x) = A(eikx – e-ikx) (V(x) = 0). Get solution
6. What are the partial wave phase shifts ... for hard-sphere scattering (Example 11.3)? ... ... That's the exact answer, but it's not terribly illuminating, so let's consider the limiting case of low-energy scattering: ..., this amounts to saying that the wavelength is much greater than the radius of the sphere.) Referring to Table 4.4, we note that ..., for small z, so ... Get solution
7. Find the S-wave (l = 0) partial wave phase shift δ0(k) for scattering from a delta-function shell (Problem 1). Assume that the ratial wave function u(r) goes to o as r →∞ Answer:...Problem 1Consider the case of low-energyscattering from a spherical delta function shell:V(r) = αδ(r-a)Where α and a are constants. Calculate the scattering amplitude, f(θ), the differenential cross-section, d(θ), and the total cross-section, σ. Assume ka « 1, so that only the l = 0 term contributes significantly. (To simplify matters, throw out all I ≠ 0 terms right from the start.) The main problem, of course, is to determine ao. Express your answer in terms of the dimensionless quantity ß ≡ 2maa/h2. Answer: σ = 4πa2ß2/(l + ß). Get solution
8. Check that Equation 11.65 satisfies Equation 11.52, by direct substitution. Hint: ... ... Get solution
9. Show that the ground state of hydrogen (Equation 4.80) satisfies the integral form of the Schrodinger equation, for the appropriate V and E (note that E is negative, so .... ... Get solution
10. Find the scattering amplitude, in the Born approximation, for soft-sphere scattering at arbitrary energy. Show that your formula reduces to Equation 11.82 in the low-energy limit. ... Get solution
11. Evaluate the integral in Equation 11.91, to confirm the expression on the right. ... Get solution
12. Calculate the total cross-section for scattering from a Yukawa potential, in the Born approximation. Express your answer as a function of E Get solution
13. For the potential in Problem 1,(a) calculate f(θ), D(θ), and σ, in the low-energy Born approximation;(b) calculate f (θ) for arbitrary energies, in the Born approximation;(c) show that your results are consistent with the answer to Problem 11.4, in the appropriate regime.Problem 1Calculate θ (as a function of the impact parameter) for Rutherford scattering, in the impulse approximation. Show that your result is consistent with the exact expression (Problem 1), in the appropriate limit.Problem 1Rutherford scattering. An incident particle of charge q1 and kinetic energy E scatters off a heavy stationary particle of charge q2.(a) Derive the formula relating the impact parameter to the scattering angle. Answer: b = (q1q2/8π∑0E) cot(θ/2). Get solution
14. Calculate θ (as a function of the impact parameter) for Rutherford scattering, in the impulse approximation. Show that your result is consistent with the exact expression (Problem 1), in the appropriate limit.Problem 1Rutherford scattering. An incident particle of charge q1 and kinetic energy E scatters off a heavy stationary particle of charge q2.(a) Derive the formula relating the impact parameter to the scattering angle. Answer: b = (q1q2/8π∑0E) cot(θ/2). Get solution
15. Find the scattering amplitude for low-energy soft-sphere scattering in the second Born approximation. Answer: —(2m Voa3/3h2)[1 - (4mVoa2/5h2)]. Get solution
16. Find the Green's function for the one-dimensional Schrodinger equation, and use it to construct the integral form (analogous to Equation 1). Answer:...Equation 1... Get solution
17. Use your result in problem 1 to develop the Born approximation for one-dimensional scattering (on the interval -∞ ...problem 1Find the Green's function for the one-dimensional Schrodinger equation, and use it to construct the integral form (analogous to Equation 1). Answer:...Equation 1... Get solution
18. Use the one-dimensional Born approximation (Problem 11.17) to compute the transmission coefficient (T = 1 — R) for scattering from a delta function (Equation 2.114) and from a finite square well (Equation 2.145). Compare your results with the exact answers (Equations 2.141 and 2.169). V (x) = -αδ(x) Get solution
19. Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude....Hint : Use Equations 1 and 2.Equation 1...Equation 2... Get solution
20. Use the Bom approximation to determine the total cross-section for scattering from a gaussian potential...Express your answer in terms of the constants A, μ, and m (the mass of the incident particle), and k ≡ ..., where E is the incident energy. Get solution
