Chapter 22 Quiz: Scattering Theory

Multiple Choice (Questions 1--14)

1. The scattering boundary condition requires that at large $r$, the wavefunction takes the form $\psi \to e^{ikz} + f(\theta)e^{ikr}/r$. The scattered wave is an outgoing spherical wave because:

(a) The Schrodinger equation only permits spherical waves as solutions (b) The scattered wave must propagate away from the scatterer, and the $1/r$ falloff conserves probability flux (c) The Born approximation requires outgoing boundary conditions (d) Ingoing spherical waves are unphysical in all circumstances

2. The differential cross section $d\sigma/d\Omega$ has dimensions of:

(a) Dimensionless (it is a probability) (b) Length (c) Area (length squared) (d) Area per steradian per second

3. In the Born approximation, the scattering amplitude is proportional to:

(a) The potential evaluated at the scattering angle (b) The Fourier transform of the potential evaluated at the momentum transfer $\mathbf{q} = \mathbf{k} - \mathbf{k}'$ (c) The second derivative of the potential at the origin (d) The expectation value of the potential in the incident state

4. The momentum transfer $q = |\mathbf{k} - \mathbf{k}'|$ for elastic scattering ($|\mathbf{k}| = |\mathbf{k}'| = k$) at angle $\theta$ is:

(a) $q = k\sin\theta$ (b) $q = k\cos(\theta/2)$ (c) $q = 2k\sin(\theta/2)$ (d) $q = 2k\cos(\theta/2)$

5. The Born approximation is most reliable when:

(a) The potential is attractive (b) The incident energy is low (c) The scattered wave is a small correction to the incident wave (weak potential or high energy) (d) Only s-wave scattering contributes

6. In the partial wave expansion, only $m = 0$ spherical harmonics (i.e., Legendre polynomials) appear because:

(a) The Legendre polynomials form a complete set (b) The problem has azimuthal symmetry when the beam is along $\hat{z}$ and the potential is central (c) Higher $m$ values correspond to unphysical solutions (d) The Born approximation selects $m = 0$

7. The phase shift $\delta_l$ for an attractive potential ($V < 0$) is typically:

(a) Positive, because the particle speeds up inside the potential and accumulates extra phase (b) Negative, because attractive potentials reduce the wavelength (c) Zero, because attractive potentials do not scatter (d) Imaginary, because the particle is absorbed

8. The maximum cross section that a single partial wave $l$ can contribute is:

(a) $\pi/k^2$ (b) $(2l+1)\pi/k^2$ (c) $4\pi(2l+1)/k^2$ (d) $4\pi l^2/k^2$

9. At low energies ($ka \ll 1$, where $a$ is the potential range), scattering is dominated by:

(a) All partial waves equally (b) The $l = 1$ (p-wave) channel (c) The $l = 0$ (s-wave) channel (d) Partial waves with $l = ka$

10. A Breit-Wigner resonance in the $l$-th partial wave occurs when:

(a) $\delta_l = 0$ (b) $\delta_l = \pi/4$ (c) $\delta_l = \pi/2$ (or odd multiple thereof) (d) $\delta_l = \pi$

11. The width $\Gamma$ of a Breit-Wigner resonance is related to the lifetime $\tau$ of the quasi-bound state by:

(a) $\Gamma = \hbar/\tau$ (b) $\Gamma = \tau/\hbar$ (c) $\Gamma = \hbar^2/\tau$ (d) $\Gamma = 2\pi\hbar/\tau$

12. The optical theorem states that $\sigma_{\text{tot}} = (4\pi/k)\operatorname{Im}[f(0)]$. This result is fundamentally a consequence of:

(a) The Born approximation (b) The reality of the potential (c) Conservation of probability (unitarity) (d) The spherical symmetry of the potential

13. The Rutherford scattering cross section $d\sigma/d\Omega \propto 1/\sin^4(\theta/2)$ has the property that:

(a) The total cross section is finite and equals $\pi a_0^2$ (b) The total cross section is infinite because the Coulomb potential has infinite range (c) The total cross section is zero because positive and negative contributions cancel (d) The total cross section depends on the nuclear spin

14. The quantum hard-sphere cross section at low energy ($ka \ll 1$) is $4\pi a^2$, which is four times the classical geometric cross section $\pi a^2$. This factor of 4 arises from:

(a) The spin-1/2 nature of the scattered particle (b) Quantum interference between incident and scattered waves (c) The Born approximation overcounting by a factor of 4 (d) Relativistic corrections to the kinetic energy

Short Answer (Questions 15--20)

15. A scattering experiment measures $d\sigma/d\Omega = 25\;\text{fm}^2/\text{sr}$ at all angles. What is the total cross section in fm$^2$ and in barns? What is the scattering amplitude $|f|$?

16. The Born approximation for a Yukawa potential gives $f(\theta) = -2mV_0/[\hbar^2(4k^2\sin^2(\theta/2) + \mu^2)]$. Explain why this amplitude is real. What does this imply about the optical theorem for the first Born approximation? Does this mean the Born approximation violates probability conservation?

17. You compute phase shifts for a certain potential and find $\delta_0 = 1.2$ rad, $\delta_1 = 0.3$ rad, $\delta_2 = 0.01$ rad, with all higher phase shifts negligible. Compute the total cross section (in units of $1/k^2$). Which partial wave dominates?

18. A resonance in the $l = 2$ partial wave is observed at energy $E_r = 3.5\;\text{MeV}$ with width $\Gamma = 0.2\;\text{MeV}$. What is the lifetime of the quasi-bound state in seconds? What is the peak cross section at resonance (in units of $1/k_r^2$)?

19. The S-matrix element for the $l = 0$ channel is $S_0 = e^{2i\delta_0}$ for elastic scattering. If $S_0 = 0.8 e^{i\pi/3}$, what is the phase shift $\delta_0$? What is the inelasticity parameter $\eta_0$? Is there absorption in this channel?

20. You are given two potentials: (A) a deep, narrow square well ($V_0 = 100$, $a = 0.5$) and (B) a shallow, wide square well ($V_0 = 1$, $a = 10$), both in atomic units. For each, state whether the Born approximation is valid at $k = 1$. Which potential is more likely to exhibit resonances? Justify your answer.