Approaching a potential from the left looks very much like what we expect to see when playing backwards a movie of what happens when we approach the potential form the right. We now use the idea of reversing time to study these two distinct processes. The only place where i appears in the TDSE is next to the time derivative operator,
Reversing time (letting 
) would simply change the
sign on the first term in this equation. We could achieve the same
effect more simply by just letting 
. Inverting the
sign on i like this is the process of complex conjugation. Thus, to
study what happens when we reverse time, we consider what happens when
we take the complex conjugate of the wave function. As we will see,
this has precisely the indented effect: doing this to a solution to
the TISE produces another valid solution, but one in which the
direction of all currents is reversed.
The reason why complex conjugated wave functions also satisfy the TISE is that the TISE is a real differential equation,
Because no factors of i appear explicitly in this equation, taking the complex conjugate of both sides gives,
We thus see that if 
is a solution
to the TISE, then so is the function 
.
Algebraically, the result of doing this to the wave function (12) is
Figure 4d shows the corresponding Feynman diagram.
Our new solution to the TISE differs from the original in two ways. First, we have had the intended effect of reversing the direction of all currents in the wave function. A subsidiary effect is that the associated quantum amplitudes are all complex conjugated as well. Although we began with the wave function for scattering from Scatter 1 from the right, inspection of the new solution which we have generated to the TISE shows that we cannot use it directly to determine the scatting amplitude for Scatterer 1 from the left. The new wave function describes an odd, but completely time reversed scenario, where particles now first approach the potential from both directions and then are transmitted out in a single beam moving to the right.
Although the new wave function which we have produced does not correspond to a familiar scenario, because the TISE is a linear equation, we may make a linear combination of the wave function (d) with the original wave function (c) and produce another solution to the TISE. By the proper choice of linear combination, we may arrange to reproduce an appropriate wave function for left-incident scattering. The appropriate combination is carried out diagrammatically in Figure 6, from which we may extract the reflection and transmission amplitudes for approaching Scatterer 1 from the left.
Figure 6: Use of time reversal symmetry to compute left- from
right-scattering amplitudes
Identifying terms from Figure 4a, we first conclude that
where we have used conservation of probability, 
.
This result is somewhat surprising. Physically, the equality of left-
and right-incident transmission amplitudes means that the
transmission probabilities and time delays for a barrier are
independent of the direction of approach. The barrier need not be
symmetric!
Of course, if the transmission probabilities are the same for both
directions, then the probabilities for reflection must be the same.
The situation for the reflection time delays a little more subtle.
Comparing Figures 4a and 6 shows that
. Rearranging and using the fact
that 
leads to 
. Analyzing the
phases in this relationship we see that
. Because the
derivatives of these phases determine the time delays, the
sum of the time delays for reflection in both direction is exactly the
same as the sum of the two time delays.
An interesting special case is that of a symmetric scatterer. In this case, the reflection delays in both directions must be equal by symmetry, but then because they give the same sum as the two equal transmission delays, the time delay for reflection and the time delay for transmission are identical for a symmetric scattering potential.