For plane wave solutions we found a phase shift upon reflection.
Is this phase shift related to some observable effect?

Let us construct a wave packet with k<k₀, incident from the left.  Let g(k)=0 for  k³k₀.  For each plane wave we have

 ,

with

.

In region 1, at t=0, we may write the wave packet as

In region 1 V=0, the wave packet represents a free particle.  For a free particle we have

and

or

with k^’=-k and g’(k^’)=g(k).  We again assume that g(k) has a pronounced peak at some k=k_i, g’(k^’) has a pronounced peak at k^’=-k_i.  Let us choose g(k) to be real, a(k)=0.  For the centers of the incident and the reflected wave packets we then have:

,

,

We have to evaluate ¶q/¶k.  We know tanq in terms of k.  With

we have

 .

Therefore

The incident and the reflected wave packet exist only in the region x<0.  For times t<<0, x_i moves with constant velocity to the right; x_r is positive and does not lie in region 1, there is no reflected wave.  For times t>>0, x_i is positive and does not lie in region 1, there is no incident wave; x_r moves with constant velocity to the left.  But the reflection has introduced a time delay T.  We have

This time delay is a direct consequence of the phase shift that we found for each plane wave solution.  The particle is not instantaneously reflected.  This is in principle a measurable effect.

Summary

When a wave packet encounters a potential step V₀ and it contains plane wave components with E>V₀, then it is partially reflected and partially transmitted.  If it contains only plane wave components with E<V₀, then it is totally reflected.  But for some time DT the wave packet penetrates into the classically forbidden region.  This results in a time delay upon reflection.

Links:

Write your own program

This link shows you how to find a numerical solution of the time-dependent Schroedinger equation in one dimension for a wave packet representing an electron confined to a region between x=0 and x=L with a potential step at x=L/2 and presents you with an example program.