18.5. Non-relativistic scattering of distinguishable particles Consider the non-relativistic Hamiltonian H =− ∂2 x

2ma

−

∂2 y

2mb

+2λδ(x−y) (18.12.33)

and let χ(x,y) be its scattering wavefunction, where the coordinate x is referred to the first particle of mass ma and y to the second particle of mass mb.

a. Show that in the centre of mass frame the Hamiltonian can be written as H =− 1 2M

∂2X

− 1 2μ

∂2Y

+2λδ(Y)

where X = max+mby ma+mb

, Y = x−y, M is the total mass M = ma +mb and μ the reduced mass μ = mamb/M.

b. Consider now the scattering process in the centre of the mass frame in which the particle of reduced mass μ is approaching the origin with ingoing planewave of momentum k. Find the transmission and reflection coefficients T and R, parameterized in terms of the velocity v = μ

−1k,

R(v) = −i2λ
v+i2λ
, T(v) = v v+i2λ
by solving the eigenstate problem with boundary conditions:
χCM(Y) = *eikY +R(v)e−ikY Y < 0 T(v)eikY Y > 0.
Notice that for any non-trivial interaction λ = 0 it is impossible to have a purely transmissive scattering, since R is always non-zero.