Wave Propagation in Linear Media | re-examined

8.4 Test of the package
8.4.2 Formal veri cation
Apart from these numerical examples, there is also a rather formal way to check the implementation.
Since the wave function is in principle a Fourier integral over the space of wave
numbers , each of the partial waves that make up the whole solution is itself a solution of
Schrodinger's equation. In addition, the partial waves on both sides of the tunnel for a given
wave number must satisfy the boundary conditions. Thus we can use the integrands instead
of the entire integrals for veri cation. Although the conditions must be met irrespective of the
wave number, we must, of course, keep in mind that we haveintroduced di erent de nitions
for di erent ranges of .
To perform the checks we must rst declare the function modules from which the integrands
are composed because they are not visible from outside the Mathematica package. As this is
quite a lengthy input, we only quote a few de nitions.
In[19]:= (*-- Functions independent of the waveform --*)
TunCoeff[xi_] := 2 xi/(xi + Sqrt[xi^2-1]);
TunOsc[a_,b_,c_,d_,xi_] :=
Exp[I (a xi^2 + b xi + c Sqrt[xi^2 - 1] + d)];
RefCoeff[xi_] := (xi - Sqrt[xi^2-1])/(xi + Sqrt[xi^2-1]);
OutOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + (b+c) xi + d)];
(*-- Triangular Shape inside the tunnel --*)
TriaConst[w_,k_] := Sqrt[3/w] / (2 k Pi^2);
TriaShapePos[w_,xi_] := 1/(1-xi/Sqrt[w])^2;
In[20]:= TriaShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w])^2;
TriaTransPos[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi];
TriaTransNeg[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi];
TriaEvan[a_,b_,c_,d_,w_,k_][xi_] :=
TriaTransPos[a,b,c,d,w,k][xi] *
(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] -
Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]);
(*-- and so on for the rest of the shapes --*)
To start with the veri cation of the boundary conditions, we rst check that the waves
inside and outside the tunnel have the same value and the same derivative at the edge. The
arguments a to d are the coe cients of the oscillatory factor. According to, for instance, (8.30)
we know that a = ,t and c = x for waves propagating in positive or negative directions,
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