Scattering of identical particles. The results for scattering of a particle from a fixed target also apply to the scattering of two particles in the center of mass frame. With Ψ(R,r) = Ψ_R (R) Ψ_r (r), Ψ_r (r) satisfies

(see Problem 5.1) where V(r) is the interaction between the particles (assumed here to depend only on their separation distance). This is the one-particle Schrödinger equation (with the reduced mass μ in place of m).
(a) Show that if the two particles are identical (spinless) bosons, then Ψr(r) must be an even function of r (Figure 10.15).
(b) By symmetrizing Equation 10.12 (why is this allowed?), show that the scattering amplitude in this case is

where f(θ) is the scattering amplitude of a single particle of mass μ from a fixed target V(r) .
(c) Show that the partial wave amplitudes of vanish for all odd powers of ℓ .
(d) How are the results of (a)–(c) different if the particles are identical fermions (in a triplet spin state).
(e) Show that the scattering amplitude for identical fermions vanishes at π/2.
(f) Plot the logarithm of the differential scattering cross section for fermions and for bosons in Rutherford scattering (Equation 10.93).

Figure 10.15: Scattering of identical particles.

Problem 5.1

Transcribed Image Text:

h 2μ V²r + V(r) v = Er Vr (10.118)