Figure 1 Barrier Potential

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

(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

(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 ACexp(+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

(1.4)

The first derivative of p is z so that (1.4) becomes

(1.5)

Multiply the top line in (1.5) by zLm(a) and add it to the bottom line

(1.6)

TI=AC*(ZLM+ZLP)*EXP(PLP)-(ZLM-ZRM)*EXP(-PRM)

Or

(1.7)
The top line of (1.5) then yields

(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 theprinciple 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

(1.9)

The value of DC can be set equal to 1 so that (1.7) becomes

(1.10)

And

(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.forILB.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