In the case 
, (28) gives t and r as just
real numbers because k and 
are real. The
phases on t and r are just 
and 
for 
(an upward step) or 
for 
(a downward
step). For either an upward or downward step, so long as , the
phases 
and 
are
independent of k and therefore, according to (10) there is
no time lag in the generation of the transmitted or reflected packets.
The case 
is somewhat different. There is no transmitted
packet propagating in the forbidden region. However, there is a
reflected packet 
which we have already studied in
Section 3.7. There we showed that
the time lag in the
emergence of the reflected packet depends on the derivative of the
phase of r(k).
Figure 12: Components of in the complex plane
To help find
this derivative we refer to Figure 12. As shown in
the figure the complex numbers 
and 
appear
opposite each other across the x-axis. Thus, if we denote
the phase angle of by 
then then phase angle of
will be 
. Now, because the phase angle of the
ratio of two complex numbers is the difference between their phase
angles and 
, we have 
.
Next we note from (29) that 
which is also indicated in the figure as the length of the hypotenuse of
each right triangle in the figure. From the figure we see that
, and thus 
.
Putting all of this together, we find
For our labor we are rewarded with a simple result with a beautiful
physical interpretation! The
packet spends an extra time 
in the forbidden
region. But, this is just the extra time the packet would take to
penetrate a distance 
into the forbidden region and then
return. And, is just the quantum penetration depth we
expect into Region t!!!