tunneling example BD 3.6.4

define wavefunctions in the three regions of space, psia for x<0
psib 0<x<a
psic x>a

psia[x_]=Exp[I k x]+r Exp[-I k x];
psib[x_]=AA Exp[κ x]+BB Exp[- κ x];
psic[x_]=t Exp[I k x];

set up boundary conditions:  wavefunctions and their derivatives must be equal at the boundaries

boundarycond={psia[0]==psib[0],psia'[0]==psib'[0],psib[a]==psic[a],psib'[a]==psic'[a]}

sol=Solve[boundarycond,{r,t,AA,BB}]//Flatten

Thad’s complex conjugate function (assumes all variables are real)

conj[a_]:=a/.Complex[x_,y_]->Complex[x,-y]

t/.sol means:  replace t by the value given in sol

TT=(t/.sol) conj[t/.sol]//Simplify

RR=(r/.sol) conj[r/.sol]//Simplify

check conservation of particles

Simplify[RR+TT]==1

or

RR+TT//Simplify

find behavior in limit of aκ>>1

TT/.Exp[ q_ a κ]->1/(b^q)

Series[%,{b,0,2}]//Normal

%/.b->Exp[-a κ]

Note, typographical error in book for Eq 3.51

Plot[TT//.{k->κ,κ->1},{a,0,10},PlotRange->All]