To sketch the outlines of the QTBM consider a one-dimensional continuum
problem defined by Schrödinger's equation with a potential 
which is constant for 
and 
(and which will in
general have different values in these two semi-infinite regions). For
a given energy E such that 
and 
, the general
solution to Schrödinger's equation 
in these regions can be
written as:
where 
and 
are the amplitudes of the incoming wave
components and 
and 
are the amplitudes of the outgoing
wave components. Now, at the left-hand boundary:
We may readily solve equation (2) for 
and similarly solve
for 
to obtain:
Equations (3) and (4) now provide the QTBM
boundary conditions, if one chooses to specify the values of 
and 
. Boundary conditions of this form are known as Robbins
conditions. They are implicit, in the sense that the values of 
,
, 
, and 
must be obtained by solving
(3) and (4) simultaneously with the
differential equation itself. If one uses a discrete approximation to
Schrödinger's equation
this presents no problem, because the differential equation is reduced
to a set of algebraic equations and the boundary conditions
(3) and (4) simply add two more equations
to this set.
