The results (21) and (22) are general for any
two scatterers. In the the special case where the two scatters are
mirror images of one another and separated by a propagating region, we
find a very beautiful phenomena known as resonance, in which at an
infinite sequence of discrete incoming energies, the transmission
probability becomes exactly one, regardless of the form of the
individual barriers. This means for instance, that although the
probability for a macroscopic object quantum tunneling through a wall
may be very low, the probability for tunneling across a double wall
becomes one for a whole series of special incoming energies. To prove
this we need only the result (21) and the general
scattering relations 
, 
from Section
3.2.
Because the t's and r's are equal, let us define
and
where 
are the probabilities for transmission and
reflection across the individual barriers (which are direction
independent), 
gives the time delay for transmission
across the barriers, and 
gives the time delay
crossing the barriers from the inside, where 
is the
appropriate classical velocity.
The transmission probability across both barriers is then
Thus, whenever 
where 
,
the transmission probability goes to one. Generally, 
does
not vary very rapidly with the incoming energy, so that this condition
is met for an infinite series of special values of 
at spacing
.
