Quantum Mechanics, Potential Step

[M. next]

I have noticed that while we study the case where, E>Vo, and we write the wave functions in both regions, we don't consider the case where x tends to infinity and thus the solution disappears.
(We typically do that to reduce unkniwn constants)

But when we consider the case where E<Vo, and write the wave functions in both regions, we do that as x tends to infinity.

Note that the difference between thojse two was that the solution in the first region, the exponentialnwas raised to a "power" containing the imaginary nb i, unlike in the second case..

Why?

 

[LagrangeEuler]

In the second case you deal with a bond states which goes to zero when $x$ goes to infinity.