Scattering theory - differential and total scattering cross sections, Born approximation, partial wave analysis and phase-shift analysis, relativistic wave equations - Klein–Gordon and Dirac equations and free-particle solutions - One Line Questions

1. When using the Born approximation for scattering, the scattering amplitude f(θ, φ) is given by: — -(m / 2πħ²) ∫ V(r') e^(i(k-k')⋅r') dr'
2. The Klein–Gordon equation is relativistically invariant and has the form: — (∂²/∂t² - ∇²)ψ - m²ψ = 0
3. The Dirac equation is a first-order differential equation in both space and time and is Lorentz invariant. It has the form: — (iγ⋅∂ - m)ψ = 0
4. What is the spin of particles described by the Dirac equation? — 1/2
5. The Dirac equation naturally accounts for the spin of the electron, which is: — 1/2
6. The free-particle solutions of the Dirac equation are represented by: — A four-component spinor.
7. The total scattering cross-section σ is obtained by integrating the differential scattering cross-section dσ/dΩ over: — The entire solid angle (4π sr).
8. The cross-section in scattering theory has units of: — Area.
9. The Dirac equation was developed to describe: — Fermions with spin 1/2, incorporating relativity.
10. The Dirac equation's success in predicting the positron was a major triumph for: — Quantum electrodynamics.
11. Which relativistic wave equation is used for spin-0 particles? — Klein–Gordon equation
12. The quantity (dσ/dΩ) relates to the scattering amplitude f(θ, φ) by: — dσ/dΩ = |f(θ, φ)|²
13. For the Klein–Gordon equation, the free-particle solutions are plane waves with energy E and momentum p satisfying: — E = ±√(p²c² + m²c⁴)
14. The Klein–Gordon equation is derived from the relativistic energy-momentum relation E² = p²c² + m²c⁴ by substituting: — E = iħ∂/∂t and p = -iħ∇
15. For a free particle described by the Klein–Gordon equation, the energy-momentum relation is: — E² = p²c² + m²c⁴
16. In scattering theory, the scattering amplitude f(θ, φ) describes the: — Amplitude of the scattered wave in a particular direction.
17. Partial wave analysis decomposes the scattering amplitude into contributions from different: — Angular momentum states (l).
18. In partial wave analysis, the scattering amplitude for s-wave (l=0) scattering is given by: — f(θ) = (1/k) e^(iδ₀) sin(δ₀)
19. In partial wave analysis, the scattering amplitude is given by the sum over all partial waves: — f(θ) = Σ (2l+1) P_l(cosθ) e^(iδ_l) sin(δ_l) / k
20. The Klein–Gordon equation describes the behavior of: — Bosons (integer spin particles).
21. The Klein–Gordon equation, despite its issues with probability, is crucial in quantum field theory for describing: — Spin-0 bosons (like the Higgs boson).
22. What is the main advantage of partial wave analysis over the Born approximation? — It is particularly effective for low-energy scattering.
23. A major problem with the Klein–Gordon equation when interpreted as a single-particle wave equation is: — It leads to negative probabilities.
24. For low-energy scattering (ka << 1, where k is the wave number and a is the range of the potential), which partial wave usually dominates? — l = 0 (s-wave)
25. The Dirac equation predicts that electrons have an intrinsic magnetic moment proportional to their: — Spin.
26. Solutions to the Dirac equation for a free particle predict the existence of: —
27. The term 'relativistic wave equation' implies that the equation is: — Invariant under Lorentz transformations.
28. What is the wave function for a free particle satisfying the Klein–Gordon equation? — Plane wave of the form e^(i(p⋅x - Et)/ħ)
29. A free particle solution to the Dirac equation can be written as a superposition of four basis states, corresponding to: — Positive energy, spin up/down and negative energy, spin up/down.
30. The probability current density for the Klein–Gordon equation has issues with: — Conservation of probability.
31. The gamma matrices (γ^μ) in the Dirac equation are: — 4x4 matrices.
32. The solution to the free-particle Dirac equation involves four-component spinors, denoted by ψ. The components represent: — Different spin and energy states.
33. The Dirac equation naturally incorporates: — Spin-1/2 particles and predicts spin.
34. What is the unit of total scattering cross-section (σ)? — Square meters (m²)
35. What is the physical interpretation of the total scattering cross-section (σ)? — The area presented by the scattering center to the incident beam for a particular scattering event.
36. The phase shift δ_l in partial wave analysis represents: — The difference in phase between the scattered wave and the incident spherical wave for a given l.
37. Phase-shift analysis is a method used to determine: — The form of the scattering potential from experimental cross-sections.
38. The Dirac equation successfully predicted the existence of: — The positron (antiparticle of the electron).
39. In the context of the Born approximation for scattering, the scattering amplitude f(θ, φ) is proportional to the Fourier transform of: — The scattering potential V(r).
40. The Born approximation is a perturbative method. The first Born approximation calculates the scattering amplitude using the incident wave function to represent the state during the scattering event. This is valid when: — The wavelength of the incident particle is much smaller than the range of the potential.
41. In phase-shift analysis, if δ_l is close to an integer multiple of π, it means: — The l-th partial wave is weakly scattered or not significantly affected.
42. The phase shift δ_l becomes significant when the incident particle's de Broglie wavelength is roughly comparable to: — The range of the potential.
43. In scattering theory, what does the differential scattering cross-section dσ/dΩ represent? — The probability of scattering into a unit solid angle.
44. The Klein–Gordon equation is a relativistic wave equation that is second-order in: — Both time and space derivatives.
45. In scattering theory, what is the primary role of the scattering potential V(r)? — To cause the incident wave to deviate from its original path.
46. The negative energy solutions of the Dirac equation were later interpreted by Dirac as: — Representing antiparticles.
47. The Born approximation is most valid when the potential V(r) is: — Weak and short-ranged.
48. The optical theorem relates the total scattering cross-section (σ_tot) to the imaginary part of the forward scattering amplitude (f(0)): — σ_tot = (4π/k) Im[f(0)]
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