Question

It is mentioned that for two classical identical particles, whose interaction potential is central the differential scattering cross section is given by (in center of mass frame), (dσ(θ)/dΩ)cl=|f(θ)|²+|f(π−θ)|² Heref(θ) is the scattering amplitude, which appears in the scattered wave function.

Then after a few lines it is mentioned that for distinguishable particles whenθ=π/2, the differential cross section is|f(π/2)|2. classical identical particles are always distinguishable. So, it should be2 |f (π/2)|2, not |f (π/2)|2, according to the 1st formula. What am are you missing here ?

Answer 1

In classical mechanics, particles are always distinguishable, leading to the differential scattering cross section being |f(π/2)|² for distinguishable particles at θ=π/2. This is due to there being only one term for the scattering amplitude, unlike the sum for identical particles. The principle of wave interference in systems like the double slit experiment relates amplitude and intensity.

Step-by-step explanation:

The differential scattering cross section you're referring to is linked to the concept of identical particles in quantum mechanics, where particles are indistinguishable and obey specific statistical rules (Bosons follow Bose-Einstein statistics, and Fermions follow Fermi-Dirac statistics). In the classical mechanics context, particles are always distinguishable, even if they are identical.

When considering the scattering of distinguishable particles, there will generally be a single term for the scattering amplitude f(θ) because one can track which particle is which throughout the interaction. This differs from the identical particle scattering, where you cannot distinguish between the two particles after scattering, which leads to the cross section being a sum of the squared amplitudes corresponding to θ and π−θ.

For distinguishable particles at θ=π/2, the cross section simplifies to |f(π/2)|² because there is no additional scattering amplitude to consider from the π−θ term, as there would be with identical particles.

When relating this to wave phenomena, such as the Young's double slit experiment, the amplitude of the overall wave pattern (A²) depends on the interference of individual waves coming from the slits, emphasizing the crucial role of phase difference (cosδ).

Maximum intensity for the central fringe, and hence the greatest probability density, occurs when the phase difference is zero. This demonstrates the direct relationship between wave amplitude and intensity of the interference pattern.