Question
Given
ˆH|ψ⟩=(12m∂2∂x21+12m∂2∂x22+k|∏ilimt→Ti1⟨H⟩ψt−Ti(x2−x1))||ψ⟩
where:
⟨H⟩ψ=⟨ψ|(12m∂2∂x21+12m∂2∂x22+k)|ψ⟩
k=limt→Ti⟨ψ|(12m∂2∂x21+12m∂2∂x22)|ψ⟩
What happens to time evolution of the wavefunction after a collision x2=x1 at Tj?
Classical Intuition
Let's say I want to model a gas of 2 particles where the gas collides at times Ti (including the collisions) -
I use the following Hamiltonian:
H=12m˙x21+12m˙x22+k∏ilimt→Tif(t−Tix2−x1)
Note: k is a parameter which obeys:
k>12m˙x21+12m˙x22
Notice at the time of a collision at when Ti→t then x2(t)−x1(t)→0.
One normalise f so that:
limt→TH(t)=limt→T12m˙x21+12m˙x22+k
Hence, the f is:
f(t−Tix2−x1)=|t−Ti(12m˙x21+12m˙x22+k)(x2−x1)|
Hence, we have:
H=12m˙x21+12m˙x22+k|∏ilimt→Ti1Ht−Ti(x2−x1)|
Hence, upon quantisation in the Schrodinger picture:
ˆH|ψ⟩=(12m∂2∂x21+12m∂2∂x22+k|∏ilimt→Ti1⟨H⟩ψt−Ti(x2−x1)|)|ψ⟩
where:
⟨H⟩ψ=limt→Ti⟨ψ|(12m∂2∂x21+12m∂2∂x22+k)|ψ⟩