Barrier
Figure 1 Barrier potential
A square well treatment of this system is in ..\1-dFlats\BarrierPotential.docx. The p used in that
treatment is such that in the region to the left of zero the wave function is
 left  x   AC exp  pL x   BC exp   pL x 
 right  x   EC exp  pR x   FC exp   pR x 
pL  i 2  E  VL 
pR  i 2  E  VR 
(1.1)
(1.2)
The middle terms have been left out of (1.1) and (1.2). The differential equation easily starts in a region
where E is greater than V and extends into a region where E is less than V, but in its present form it
cannot go from a region where E is much less than V into a region where E is greater than V. The x’s in
equation (1.1) get incorporated into pp(x) and pm(x) so that the equations become
 left  x   AC exp  pLp  x    BC exp   pLm  x  
 right  x   CC exp  pRp  x    DC exp   pRm  x  
(1.3)
Figure 2 the z values; 13 is the imaginary parts.
X'S FOR E=POT(X)
4.5000000000000000
5.0262214620764958
6.9737785379235042
The differential equation from -24 goes past the first E=V point, 5.03, with no problems, but shortly past
the second, 6.97, gives up. Going down from 24, it passes 6.97 and finally gives up slightly past 5.03.
The three complex values of z plotted here produce the initial incoming wave, 4p5L-m, the reflected
wave, 4p5L-p, and the transmitted wave, 4p5R-m.
In the approximation used here, the wave function is exp(±p(x)). The function z(x) is dp(x)/dx. The
differential equation can cross the boundary from E > V to E < V, but not the other way around. The
sokution to the differential equation is very dependent on the initial value of z. Beyond this the function
is fixed. The solution ends when the predictor corrector fails. The functions with L start from the left,
those with R from the right. The last –m or –p refers to solutions for the -p(x) or the +p(x). The
imaginary parts of ±p are the same, but the real parts are not.
Figure 3 Barrier values of p for E=3.5 where V3 has a maximum of 5
The function p(x) is the integral of z(x) starting from an arbitrary point. The wave function is AC
exp(+p(x))+BCexp(-p(x)). In consfind.for these constants are written as
ALC=LOG(AC) line 57
This allows ALC to be directly added to p(x). This means that the lines in Figure 3 can be move vertically
while exp(ALC +px) and exp(ALC -px) remain solution to the Schrodinger equation.
In general these solutions are valid from the left – top line – and the right –bottom line – up to the mid
point of the barrier at 6 in Figure 1. The solution for CC and DC in terms of AC and BC for regions with flat
potentials is described in ..\1-dFlats\FlatEdge.docx .pdf . Modifications are needed for (1.3), but for
barrier penetration, AC and BC are the unknowns while CC and DC are known.
Equate  and its derivatives at x=a
AC exp  pLp  a    BC exp   pLm  a    CC exp  pRp  a    DC exp   pRm  a  
 CC p 'Rp  a  exp  pRp  a  
 (1.4)


p
'
a
exp
p
a

B
p
'
a
exp

p
a













C
Lp
Lp
C
Lm
Lm
  DC p 'Rm  a  exp   pRm  a   


A

The first derivative of p is z so that (1.4) becomes
AC exp  pLp  a    BC exp   pLm  a    CC exp  pRp  a    DC exp   pRm  a  

 CC z Rp  a  exp  pRp  a  
 (1.5)

AC z Lp  a  exp  pLp  a    BC z Lm  a  exp   pLm  a    
  DC z Rm  a  exp   pRm  a   



Multiply the top line in (1.5) by zLm(a) and add it to the bottom line
AC  zLm  a   z Lp  a   exp  pLp  a    CC  z Lm  a   z Rp  a   exp  pRp  a    DC  z Lm  a   z Rm  a   exp   pRm  a  
(1.6)
TI=AC*(ZLM+ZLP)*EXP(PLP)-(ZLM-ZRM)*EXP(-PRM)
Or
AC 
CC  zLm  a   z Rp  a   exp  pRp  a   pLp  a    DC  z Lm  a   z Rm  a   exp   pRm  a   pLp  a  
 z  a   z  a 
Lm
Lp
(1.7)
The top line of (1.5) then yields
BC  CC exp  pRp  a   pLm  a    DC exp   pRm  a   pLm  a    AC exp  pLp  a   pLm  a  
(1.8)
CEXP2=EXP(PLP-PLM)
BC=(DC*CEXP3-AC*CEXP2)
The presence of zm(x) and zp(x) in (1.7)rather than a single p’(x) is the principle difference between this
and
..\1-dFlats\FlatEdge.docx .pdf .
This comes about because the second derivative of p(x) is not zero so that the two independent wave
functions are p and m rather than exp(p) and exp(-p). The case 2 section of ..\DiffEq\EulerExt.for
starts with purely imaginary p values at the beginning and end of the region which allows us to call
BCexp(-pLm(x)) the incoming wave and CCexp(-pRm(x)) the outgoing wave. The scattering result is
explained in
..\..\Bohm14.docx .htm .
Setting CC to zero implies that there is no incoming wave from the right. This makes the transmission
coefficient equal to
T
DC
Bc
2
2
(1.9)
The value of DC can be set equal to 1 so that (1.7) becomes
AC 
 z  a   z  a   exp   p  a   p  a  
Lm
Rm
Rm
 z  a   z  a 
Lm
Lp
(1.10)
Lp
And
BC  exp   pRm  a   pLm  a    AC exp  pLp  a   pLm  a   (1.11)
The direction file EulerExt.dir in combination with EulerExt.bat produces the files needed for a single
energy.
The code is ConsFind.wpj (consfind.for ILB.for ..\..\..\interpolation\xyfilefun.for)
..\DiffEq\EulerExtt.for used with “nameL” puts the values p’Lm(x) into the file z-nameLm.out and the
values of exp(-pLm(x)) into the file Psdi-nameLm.out with similar placements for the nameR files. This
means that the values of z and require interpolation in these files. ..\..\..\interpolation\xyfile.docx
Figure 4 Real and imaginary parts of total wave function for E=4.5
The real parts of Psi-7p5L-m and Psi-7p5L-p are on top of each other. The wave function to the left is
not of the form exp(p(x)). It is exp(pp(x))+exp(-pm(x)). The function to the right by the transmission
assumption is of the form exp(-pm(x)). The incoming wave is largest just before the top of the barrier.
EulerExt.dir
Figure 5 z for E=7.5 above the barrier peak at 5
The one from the right is smooth down to the barrier and wobbles below it. The one from the left does
the opposite.
Figure 6 m for exp(-p); p for exp(p), where p= z(x)dx
The two curves differ only in the real part of z which is non zero where the potential peaks.
Figure 7 The total wave function
T
0.9999989854498604
The smoothness comes from joining the “good” left solution to the “good” right solution at 6.
Figure 8 Reflected wave
The reflected wave is part of the total wave in Figure 7. The sudden stop at 5 is an artifact of our
solution method.
T 1.2658377951606243D-011
Figure 9 Barrier penetration