ABSTRACT
In many experimental situations, we are interested in the scattering of two particles with momenta k 1 and k 2 to a state with n particles with momenta https://www.w3.org/1998/Math/MathML"> p 1 , p 2 , … , p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_543.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We denote by https://www.w3.org/1998/Math/MathML"> ∫ d 3 k → 1 Ψ ˜ 1 ( k → 1 ) | k → 1 > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_544.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∫ d 3 k → 2 Ψ ˜ 2 ( k → 2 ) | k → 2 > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_545.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the wave functionals of the incoming particles. This is to describe the more realistic case of a wave packet. The amplitude for scattering to take place is hence given by () https://www.w3.org/1998/Math/MathML"> A = − i ( 2 π ) 4 ∫ d 3 k → 1 d 3 k → 2 δ 4 ( ∑ i = 1 n p i − k 1 − k 2 ) M 2 + n ( { − p i } , { k j } ) ∏ i = 1 n 2 p 0 ( i ) ( p → i ) ( 2 π ) 3 ∏ j = 1 2 2 k 0 ( j ) ( k → j ) ( 2 π ) 3 × Ψ ˜ 1 ( k → 1 ) Ψ ˜ ( k → 2 ) e − i [ E 0 + ∑ p 0 ( i ) ] T . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_546.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
