Path Integral for Motion in the Quantum Well

However, there are also two characteristics, due to the collisions and the turning points along the path on the walls, namely  

Combining these information, as previously for the Bohr s atom, with the sum over infinite histories with the same scenario and output, while performing for each of the exponentials of Eq. (4.157) the same analytical transformation by means of Eq. (4.138) as previously done for atomic circular motion, we adapt the result (4.142) to the present quantum well situation to firstly get 

This way, the propagator of the particle within the quantum well further writes from Eq. (4.160) under delta-Dirac form 

Yielding upon performing the integration by the filtration property of delta-Dirac function, see Eq. (2.11), to leave with the Green function result 

Where we have recorded the energy and wave vector quantifications, respectively 

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