1. The propagator of a free particle is given by K_0 (x,t│x^',t^...

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1. The propagator of a free particle is given by K_0 (x,t│x^',t^'=0)=√(m/2πiℏt) exp⁡[(-m(x-x^' )^2)/2iℏt]. In the presence of a perfectly reflecting wall (at the origin x=0) the propagator reads: K(x,t│,x^',t^' )=K_0 (x,t│x^',t^' )-K_0 (-x,t|x^',t^'). Find the long-time behavior of the survival probability for an arbitrary initial state psi(x’,t’=0) which is initially confined in the interval [0,L] and evolves for t>0 in the absence of the wall in the presence of the wall Compare the two results. As a hint, consider the power-series expansion of the corresponding propagators as t->infinity (and setting t’=0). (c ) Verify that the same scaling behavior in (a) and (b) holds for the nonescape probability.

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