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Chapter 4. Particle scattering
where G(x; x
� ) is the Green’s function (4.64). This equation represents the main result of this section. It can be regarded as completely equivalent to the differential equation for u(x) (4.57), which
is the spatial part of Schr¨ odinger’s equation for the stationarystate case. We recall that |u(x)|
2 is the probability density that a
single, precise measurement of the scattered particle position will
find the particle at position x. This represents the connection with
experimental measurement. The equation (4.70) has the advantage of being more directly applicable to the scattering problem.
We will make use of this in the following section.
4.4 Green’s function solution for elastic scattering
We now turn our attention to the important special case of twoparticle scattering in which the incident particle transfers negligible energy to the target particle. This process is known as elastic
scattering. The central problem is to calculate the differential cross
section σ(ϑ), which gives the scattered intensity at angle ϑ.
In the case where the incident and target particles each retain
their same rest mass in the initial and final states, and we disregard internal degrees of freedom for each particle, the total energy
is conserved for each particle individually in the CM frame. In
the nonrelativistic limit, the two-body scattering is reduced to the
equivalent one-body scattering. In this case a single particle with
reduced mass M scatters from a fixed center, where M is given by
(4.15)
m 1 m 2
M =
,
(4.71)
m 1 + m 2
where m 1 and m 2 are the rest masses of the incident and target
particles, respectively.
Chapter 4. Particle scattering
where G(x; x
� ) is the Green’s function (4.64). This equation represents the main result of this section. It can be regarded as completely equivalent to the differential equation for u(x) (4.57), which
is the spatial part of Schr¨ odinger’s equation for the stationarystate case. We recall that |u(x)|
2 is the probability density that a
single, precise measurement of the scattered particle position will
find the particle at position x. This represents the connection with
experimental measurement. The equation (4.70) has the advantage of being more directly applicable to the scattering problem.
We will make use of this in the following section.
4.4 Green’s function solution for elastic scattering
We now turn our attention to the important special case of twoparticle scattering in which the incident particle transfers negligible energy to the target particle. This process is known as elastic
scattering. The central problem is to calculate the differential cross
section σ(ϑ), which gives the scattered intensity at angle ϑ.
In the case where the incident and target particles each retain
their same rest mass in the initial and final states, and we disregard internal degrees of freedom for each particle, the total energy
is conserved for each particle individually in the CM frame. In
the nonrelativistic limit, the two-body scattering is reduced to the
equivalent one-body scattering. In this case a single particle with
reduced mass M scatters from a fixed center, where M is given by
(4.15)
m 1 m 2
M =
,
(4.71)
m 1 + m 2
where m 1 and m 2 are the rest masses of the incident and target
particles, respectively.