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  A toy model of an almost interacting gas?

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Question

Given

ˆH|ψ=(12m2x21+12m2x22+k|ilimtTi1HψtTi(x2x1))||ψ

where:

Hψ=ψ|(12m2x21+12m2x22+k)|ψ

k=limtTiψ|(12m2x21+12m2x22)|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+kilimtTif(tTix2x1)

Note: k is a parameter which obeys:

k>12m˙x21+12m˙x22

Notice at the time of a collision at when Tit then x2(t)x1(t)0.

One normalise f so that:

limtTH(t)=limtT12m˙x21+12m˙x22+k

Hence, the f is:

f(tTix2x1)=|tTi(12m˙x21+12m˙x22+k)(x2x1)|


Hence, we have:

H=12m˙x21+12m˙x22+k|ilimtTi1HtTi(x2x1)|

Hence, upon quantisation in the Schrodinger picture:

ˆH|ψ=(12m2x21+12m2x22+k|ilimtTi1HψtTi(x2x1)|)|ψ

where:

Hψ=limtTiψ|(12m2x21+12m2x22+k)|ψ

asked Nov 4, 2019 in quant-ph by Asaint (90 points) [ no revision ]

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