TRB PG Assistant - 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 Online Test (MCQ Quiz)

1. In scattering theory, what does the differential scattering cross-section dσ/dΩ represent?

A) The total probability of scattering per unit time. B) The probability of scattering into a unit solid angle. C) The total number of particles scattered. D) The average kinetic energy of scattered particles.

2. What is the unit of total scattering cross-section (σ)?

A) Steradians (sr) B) Meters (m) C) Square meters (m²) D) Per second (s⁻¹)

3. The Born approximation is most valid when the potential V(r) is:

A) Very strong and long-ranged. B) Weak and short-ranged. C) Strong and short-ranged. D) Weak and long-ranged.

4. In the context of the Born approximation for scattering, the scattering amplitude f(θ, φ) is proportional to the Fourier transform of:

A) The incident wave function. B) The scattered wave function. C) The scattering potential V(r). D) The energy of the incident particle.

5. What is the physical interpretation of the total scattering cross-section (σ)?

A) The area presented by the scattering center to the incident beam for a particular scattering event. B) The total area of the target nucleus. C) The area of the detector. D) The average distance traveled by a scattered particle.

6. Partial wave analysis decomposes the scattering amplitude into contributions from different:

A) Energy states. B) Angular momentum states (l). C) Spin states. D) Particle types.

7. The phase shift δ_l in partial wave analysis represents:

A) The energy shift of the l-th partial wave. B) The change in the angular momentum of the scattered particle. C) The difference in phase between the scattered wave and the incident spherical wave for a given l. D) The probability of scattering in the l-th partial wave.

8. 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?

A) l = 1 B) l = 2 C) l = 0 (s-wave) D) All partial waves contribute equally.

9. The optical theorem relates the total scattering cross-section (σ_tot) to the imaginary part of the forward scattering amplitude (f(0)):

A) σ_tot = (4π/k) Im[f(0)] B) σ_tot = (k/4π) Im[f(0)] C) σ_tot = 4πk Im[f(0)] D) σ_tot = Im[f(0)] / 4πk

10. Phase-shift analysis is a method used to determine:

A) The form of the scattering potential from experimental cross-sections. B) The energy levels of bound states. C) The momentum of the incident particles. D) The spin of the target.

11. The Klein–Gordon equation describes the behavior of:

A) Fermions (spin-1/2 particles). B) Bosons (integer spin particles). C) Photons. D) Neutrinos.

12. The Klein–Gordon equation is relativistically invariant and has the form:

A) (∂²/∂t² - ∇²)ψ + m²ψ = 0 B) (∂²/∂t² + ∇²)ψ - m²ψ = 0 C) (∂²/∂t² - ∇²)ψ - m²ψ = 0 D) (∂²/∂t² + ∇²)ψ + m²ψ = 0

13. A major problem with the Klein–Gordon equation when interpreted as a single-particle wave equation is:

A) It predicts negative energies. B) It leads to negative probabilities. C) It is not Lorentz invariant. D) It cannot describe spin.

14. The Dirac equation was developed to describe:

A) Bosons with spin 0. B) Fermions with spin 1/2, incorporating relativity. C) Massless particles like photons. D) Composite particles.

15. The Dirac equation is a first-order differential equation in both space and time and is Lorentz invariant. It has the form:

A) (iγ⋅∂ - m)ψ = 0 B) (iγ⋅∂ + m)ψ = 0 C) (γ⋅∂ - m)ψ = 0 D) (i∂/∂t - iγ⋅∇ - m)ψ = 0

16. The gamma matrices (γ^μ) in the Dirac equation are:

A) Scalar matrices. B) 2x2 matrices. C) 4x4 matrices. D) Infinite dimensional matrices.

17. The Dirac equation naturally incorporates:

A) Spin-0 particles. B) Spin-1/2 particles and predicts spin. C) Spin-1 particles. D) Bosonic statistics.

18. Solutions to the Dirac equation for a free particle predict the existence of:

A) Only positive energy states. B) Both positive and negative energy states. C) Only zero energy states. D) Virtual particles.

19. The negative energy solutions of the Dirac equation were later interpreted by Dirac as:

A) Unphysical and should be discarded. B) Representing antiparticles. C) Related to nuclear forces. D) Quantum fluctuations.

20. What is the wave function for a free particle satisfying the Klein–Gordon equation?

A) Plane wave of the form e^(i(p⋅x - Et)/ħ) B) Spherical wave of the form e^(ikr)/r C) Gaussian wave packet. D) Superposition of plane waves.

21. For a free particle described by the Klein–Gordon equation, the energy-momentum relation is:

A) E² = p²c² + m²c⁴ B) E = pc C) E² = p²c² D) E = mc²

22. What is the spin of particles described by the Dirac equation?

A) 0 B) 1/2 C) 1 D) 3/2

23. The free-particle solutions of the Dirac equation are represented by:

A) A single scalar function. B) A two-component spinor. C) A four-component spinor. D) A vector field.

24. The Klein–Gordon equation is a relativistic wave equation that is second-order in:

A) Time derivatives only. B) Space derivatives only. C) Both time and space derivatives. D) Neither time nor space derivatives.

25. The Dirac equation successfully predicted the existence of:

A) The Higgs boson. B) The positron (antiparticle of the electron). C) Quarks. D) Neutrinos.

26. In scattering theory, the scattering amplitude f(θ, φ) describes the:

A) Energy of the scattered particle. B) Direction of the scattered particle. C) Amplitude of the scattered wave in a particular direction. D) Total cross-section.

27. 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:

A) The interaction is very strong. B) The wavelength of the incident particle is much smaller than the range of the potential. C) The kinetic energy of the incident particle is much smaller than the potential energy. D) The potential is attractive.

28. In partial wave analysis, the scattering amplitude is given by the sum over all partial waves:

A) f(θ) = Σ (2l+1) P_l(cosθ) e^(iδ_l) sin(δ_l) / k B) f(θ) = Σ (2l+1) P_l(cosθ) sin(δ_l) / k C) f(θ) = Σ (2l+1) P_l(cosθ) e^(iδ_l) / k D) f(θ) = Σ (2l+1) P_l(cosθ) e^(iδ_l) cos(δ_l) / k

29. The quantity (dσ/dΩ) relates to the scattering amplitude f(θ, φ) by:

A) dσ/dΩ = |f(θ, φ)|² B) dσ/dΩ = k |f(θ, φ)| C) dσ/dΩ = |f(θ, φ)| D) dσ/dΩ = k² |f(θ, φ)|

30. Which relativistic wave equation is used for spin-0 particles?

A) Dirac equation B) Klein–Gordon equation C) Schrödinger equation D) Proca equation

31. The probability current density for the Klein–Gordon equation has issues with:

A) Relativistic invariance. B) Conservation of probability. C) Spin dependence. D) Charge conjugation.

32. The Dirac equation naturally accounts for the spin of the electron, which is:

A) 0 B) 1/2 C) 1 D) 2

33. A free particle solution to the Dirac equation can be written as a superposition of four basis states, corresponding to:

A) Positive energy, spin up/down and negative energy, spin up/down. B) Positive energy, spin left/right and negative energy, spin left/right. C) Zero energy, spin up/down and infinite energy, spin up/down. D) Momentum up/down and energy up/down.

34. The cross-section in scattering theory has units of:

A) Angular momentum. B) Energy. C) Area. D) Probability.

35. What is the main advantage of partial wave analysis over the Born approximation?

A) It is simpler to calculate. B) It is valid for all energies and potentials. C) It is particularly effective for low-energy scattering. D) It directly gives the potential from the phase shifts.

36. In phase-shift analysis, if δ_l is close to an integer multiple of π, it means:

A) The l-th partial wave is strongly scattered. B) The l-th partial wave is weakly scattered or not significantly affected. C) Resonance occurs for the l-th partial wave. D) The scattering is purely elastic.

37. The Klein–Gordon equation is derived from the relativistic energy-momentum relation E² = p²c² + m²c⁴ by substituting:

A) E = iħ∂/∂t and p = -iħ∇ B) E = -iħ∂/∂t and p = iħ∇ C) E = ħ∂/∂t and p = ħ∇ D) E = iħ∂/∂t and p = ħ∇

38. The Dirac equation predicts that electrons have an intrinsic magnetic moment proportional to their:

A) Mass. B) Charge. C) Spin. D) Momentum.

39. The solution to the free-particle Dirac equation involves four-component spinors, denoted by ψ. The components represent:

A) Spatial coordinates. B) Time and energy. C) Different spin and energy states. D) Particle and antiparticle wave functions.

40. When using the Born approximation for scattering, the scattering amplitude f(θ, φ) is given by:

A) -(m / 2πħ²) ∫ V(r') e^(-ik'⋅r') e^(ik⋅r') dr' B) -(m / 2πħ²) ∫ V(r') e^(i(k-k')⋅r') dr' C) -(m / 2πħ²) ∫ V(r') e^(-i(k-k')⋅r') dr' D) -(m / 2πħ) ∫ V(r') e^(i(k-k')⋅r') dr'

41. The total scattering cross-section σ is obtained by integrating the differential scattering cross-section dσ/dΩ over:

A) All possible energies. B) All possible momenta. C) The entire solid angle (4π sr). D) The forward direction (θ=0).

42. In partial wave analysis, the scattering amplitude for s-wave (l=0) scattering is given by:

A) f(θ) = (1/k) e^(iδ₀) sin(δ₀) B) f(θ) = (1/k) sin(δ₀) C) f(θ) = (1/k) e^(iδ₀) D) f(θ) = e^(iδ₀) sin(δ₀)

43. The phase shift δ_l becomes significant when the incident particle's de Broglie wavelength is roughly comparable to:

A) The speed of light. B) The mass of the particle. C) The range of the potential. D) The total cross-section.

44. The Klein–Gordon equation, despite its issues with probability, is crucial in quantum field theory for describing:

A) Fermions. B) Massless vector bosons. C) Spin-0 bosons (like the Higgs boson). D) Spin-1/2 fermions.

45. The term 'relativistic wave equation' implies that the equation is:

A) Only valid for particles moving at speeds close to light. B) Invariant under Lorentz transformations. C) Derived from classical mechanics. D) Only applicable in the absence of potential fields.

46. The Dirac equation's success in predicting the positron was a major triumph for:

A) Classical electromagnetism. B) Quantum electrodynamics. C) General relativity. D) Newtonian mechanics.

47. For the Klein–Gordon equation, the free-particle solutions are plane waves with energy E and momentum p satisfying:

A) E = √(p²c² + m²c⁴) B) E = ±√(p²c² + m²c⁴) C) E = pc D) E = mc²

48. In scattering theory, what is the primary role of the scattering potential V(r)?

A) To define the initial state of the particle. B) To cause the incident wave to deviate from its original path. C) To measure the energy of the scattered particles. D) To determine the detector efficiency.

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 - Online Test

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