The SchrGdinger equations for the different regions are
![](https://www.sarthaks.com/?qa=blob&qa_blobid=12449392040213225991)
(a) E < 0.
(i) Consider first the case of KJ < -E, for which the wave function is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=6768630605833537598)
The continuity conditions of the wave function give
![](https://www.sarthaks.com/?qa=blob&qa_blobid=16680256783515009637)
As cot h x > 0 for x > 0, there is no solution for this case.
(ii) For V0 > -E, ik → k, k = 2m(V0 + E)/h, and the equation determining the energy becomes k cot (ka) = -k' The wave function is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=14889928487032077003)
From the continuity and normalization of the wave function we get
![](https://www.sarthaks.com/?qa=blob&qa_blobid=8226481968518246363)
(b) E > 0. The wave function is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=9944867219180273757)
Hence the phase shift of the outgoing wave in relation to the incident wave is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=9285463293651619939)