Question

Use appropriate connection formulas to analyze the problem of scattering from a barrier with sloping walls (Figurea).

Hint: Begin by writing the WKB wave function in the form

$\mathbf{\psi }\left(x\right)\mathbf{=}\{\begin{array}{l}\frac{1}{\sqrt{p\left(x\right)}}\left[{\mathrm{Ae}}^{\frac{i}{h}{\int }_x^{x^1}p\left(x^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}+{\mathrm{Be}}^{\frac{-i}{h}{\int }_x^{r_1}p\left(x^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}\right],\left(x<x_1\right)\\ \frac{1}{\sqrt{\left|p\left(x\right)\right|}}\left[{\mathrm{Ce}}^{\frac{i}{h}{\int }_{x_1}^{\text{'}}\left|p\left(x^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}+{\mathrm{De}}^{-\frac{1}{h}{\int }_{x_1}^X\left|p\left(x^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}\right],\left(X_1<X<X_2\right)\\ \frac{1}{\sqrt{p\left(x\right)}}\left[{\mathrm{Fe}}^{\frac{i}{h}{\int }_{x_2}^{x}p\left(x^{\text{'}}\right)\mathrm{dx}}\right].\left(x>x_2\right)\end{array}$

Do not assume C=0 . Calculate the tunneling probability, $\mathbf{T}\mathbf{=}{\left|F\right|}^{\mathbf{2}}\mathbf{/}{\left|A\right|}^{\mathbf{2}}$, and show that your result reduces to Equation 8.22 in the case of a broad, high barrier.

Short answer

The Tunneling probability is$\mathrm{T}\approx {\mathrm{e}}^{-2\mathrm{\gamma }}$.

Step-by-step solution

To find patching wave function.

We use a patching wave function $\left({\mathrm{\psi }}_{\mathrm{p}}\right)$to match the WKB wave functions on either side of the turning points.

Consider the turning point $x_1$(with upward slope).

Let's shift the axes over so that the left hand turning point $x_1$occurs at $x=0$ .

The WKB wave function takes the form

role="math" localid="1658387748450" $\psi \left(x\right)\left\{\begin{array}{l}\begin{array}{l}\frac{1}{\sqrt{\mathrm{p}\left(\mathrm{x}\right)}}[{\mathrm{Ae}}^{-\frac{\mathrm{i}}{\mathrm{h}}{\int }_{\mathrm{x}}^0\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}+{\mathrm{Be}}^{\frac{\mathrm{i}}{\mathrm{h}}{\int }_{\mathrm{x}}^0\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}],\mathrm{x}<0,\\ \frac{1}{\sqrt{\left|\mathrm{p}\left(\mathrm{x}\right)\right|}}[{\mathrm{Ce}}^{\frac{\mathrm{i}}{\mathrm{h}}{\int }_0^{\mathrm{x}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}+{\mathrm{De}}^{-\frac{1}{\mathrm{h}}{\int }_0^{\mathrm{X}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}],\mathrm{x}>0.\end{array}\\ \end{array}\right.\dots \dots \dots \dots \dots .\left(1\right)$

and the patching wave function is

${\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\mathrm{aAi}\left(\mathrm{\alpha x}\right)+\mathrm{bBi}\left(\mathrm{\alpha x}\right),\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots ..\left(2\right)$

For appropriate constants a and b .

Also, the linear approximation of the potential in the overlapping regions yields

$\mathrm{p}\left(\mathrm{x}\right)\approx \sqrt{2\mathrm{m}\left(\mathrm{E}-\mathrm{E}-{\mathrm{V}}^{,}\left(0\right)\mathrm{x}\right)}={\mathrm{h\alpha }}^{3/2}\sqrt{-\mathrm{x}}$

In particular, in overlap region 2,

${\int }_0^{\mathrm{x}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}\approx {\mathrm{h\alpha }}^{3/2}{\int }_0^{\mathrm{x}}\sqrt{{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}}=\frac{2}{3}\mathrm{h}{\left(\mathrm{\alpha X}\right)}^{3/2}$

Therefore the WKB wave function (Equation (1)) can be written as

$\mathrm{\psi }\left(\mathrm{x}\right)\approx \frac{1}{\sqrt{\mathrm{h}}{\mathrm{\alpha }}^{3/4}{\mathrm{\chi }}^{1/4}}\left[{\mathrm{Ce}}^{\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}+{\mathrm{De}}^{-\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}\right]$

Meanwhile, using the large- z asymptotic forms of the Airy functions, the patching wave function (Equation(2)) in overlap region 2 becomes,

${\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)\approx \frac{\mathrm{a}}{2\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}{\mathrm{e}}^{-\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}+\frac{\mathrm{b}}{\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}{\mathrm{e}}^{\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}$

Comparing the two solutions, we see that

$\mathrm{a}=\sqrt{\frac{4\mathrm{\pi }}{\mathrm{h\alpha }}}\mathrm{D},\mathrm{And}\mathrm{b}=\sqrt{\frac{\mathrm{\pi }}{\mathrm{h\alpha }}}\mathrm{C}$ ……………………………(3)

To derive the overlap region.

Now we go back and repeat the procedure for overlap region 1. Once again, p(x) is given by Equation9.39, but this time is x negative, so

${\text{∫}}_x^{0}p\left(x^{\text{'}}\right)d{x}^{\text{'}}\approx h{\alpha }^{3/2}{\int }_x^0\sqrt{-x^{\text{'}}}d{x}^{\text{'}}=\frac{2}{3}h{\left(-\alpha x\right)}^{3/2}$

Therefore the WKB wave function (Equation (1)) can be written as

$\mathrm{\psi }\left(\mathrm{x}\right)\approx \frac{1}{\sqrt{\mathrm{h}}{\mathrm{\alpha }}^{3/4}{\left(-\mathrm{x}\right)}^{1/4}}\left[{\mathrm{Ae}}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}+{\mathrm{Be}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}\right]$

Meanwhile, using the large- asymptotic form of the Airy function for large negative z, the patching wave function (Equation (2)) in overlap region 1 becomes

$$\begin{gathered} {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)\approx \frac{\mathrm{a}}{\sqrt{\mathrm{\pi }}{\left(-\mathrm{\alpha x}\right)}^{1/4}}\sin \left[\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}+\frac{\mathrm{\pi }}{4}\right]+\frac{\mathrm{b}}{\sqrt{\mathrm{\pi }}{\left(-\mathrm{\alpha x}\right)}^{1/4}}\cos \left[\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}+\frac{\mathrm{\pi }}{4}\right] \\ {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{\pi }}{\left(-\mathrm{\alpha x}\right)}^{1/4}}\left[-{\mathrm{iae}}^{\mathrm{c}}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}+-{\mathrm{iae}}^{-\mathrm{i\pi }//4}{\mathrm{e}}^{-\mathrm{c}}+{\mathrm{be}}^{\mathrm{i\pi }/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}+{\mathrm{be}}^{-\mathrm{i\pi }//4}{\mathrm{e}}^{{}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}}\right] \\ {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{\pi }}{\left(-\mathrm{\alpha x}\right)}^{1/4}}\left[{\mathrm{e}}^{\mathrm{i\pi }//4}\left(\mathrm{b}-\mathrm{ia}\right){\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}+{\mathrm{e}}^{\mathrm{i\pi }//4}\left(\mathrm{b}+\mathrm{ia}\right){\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}\right] \end{gathered}$$

Comparing the WKB and patching wave functions in overlap region 1,

We find

$\frac{\mathrm{b}-\mathrm{ia}}{2\sqrt{\mathrm{\pi }}}{\mathrm{e}}^{\mathrm{i\pi }/4}=\frac{\mathrm{B}}{\sqrt{\mathrm{h\alpha }}}\mathrm{and}\frac{\mathrm{b}+\mathrm{ia}}{2\sqrt{\mathrm{\pi }}}{\mathrm{e}}^{-\mathrm{i\pi }/4}=\frac{\mathrm{A}}{\sqrt{\mathrm{h\alpha }}}$

or, putting in Equation (3) for a and b :

$\mathrm{B}=\left(\frac{\mathrm{C}}{2}-\mathrm{iD}\right){\mathrm{e}}^{\mathrm{i\pi }/4}\mathrm{and}\mathrm{A}=\left(\frac{\mathrm{C}}{2}+\mathrm{iD}\right){\mathrm{e}}^{-\mathrm{i\pi }/4}$ ………………………(4)

These are the connection formulas, joining the WKB solutions at either side of the turning point x1 .

To write upward slope and downward slope.

Now we consider the turning points x2 (with downward slope) and apply the same procedure. First we need to rewrite the WKB solution over the region $\mathrm{x}1<\mathrm{x}<\mathrm{x}2$:

$\mathrm{\psi }\left(\mathrm{x}\right)\approx \frac{1}{\sqrt{\left|\mathrm{p}\left(\mathrm{x}\right)\right|}}\mathrm{Ce}\frac{1}{\mathrm{h}}\int {\mathrm{x}}_1{\mathrm{x}}_2\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}-{\int }_{\mathrm{x}}^{{\mathrm{x}}_2}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}+{\mathrm{De}}^{\frac{1}{\mathrm{h}}\left[-{\int }_{{\mathrm{x}}_1}^{{\mathrm{x}}_2}\left|\mathrm{p}{\left(\mathrm{x}\right)}^{\text{'}}\right|\mathrm{dx}+{\int }_{\mathrm{x}}^{{\mathrm{x}}_2}\left|\mathrm{p}{\left(\mathrm{x}\right)}^{\text{'}}\right|\mathrm{dx}\right]}$

Letting $\mathrm{C}\equiv {\mathrm{De}}^{-\mathrm{\gamma }},{\mathrm{D}}^{\text{'}}\equiv {\mathrm{Ce}}^{\mathrm{\gamma }},$where $\mathrm{\gamma }\equiv \frac{1}{\mathrm{h}}{\int }_{{\mathrm{x}}_1}^{{\mathrm{x}}_2}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|\mathrm{dx}$is constant (using the same notation as in Equation 9.23), the WKB wave function takes the form (after shifting the origin to $x_2$).

$\mathrm{f}\left(\mathrm{x}\right)\approx \left\{\begin{array}{l}\frac{1}{\sqrt{\mathrm{Ip}\left(\mathrm{x}\right)}\mathrm{I}}\left[{\mathrm{C}}^{\text{'}}{\mathrm{e}}^{\frac{1}{\mathrm{h}}{\int }_{\mathrm{x}}^0\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}+{\mathrm{De}}^{-\frac{1}{\mathrm{h}}{\int }_{\mathrm{x}}^0\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}\right],\mathrm{x}<0\\ \\ \frac{1}{\sqrt{\mathrm{p}\left(\mathrm{x}\right)}}\left[{\mathrm{Fe}}^{\frac{\mathrm{i}}{\mathrm{h}}{\int }_{\mathrm{x}}^0\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}\right],\mathrm{x}>0\end{array}\right.$ …………………..(5)

In this case, the linear approximation of the potential is

$\mathrm{V}\left(\mathrm{x}\right)\approx \mathrm{E}-{\mathrm{V}}^{\text{'}}\left(\mathrm{o}\right)\mathrm{x}$

Accordingly, the patching wave function$\left({\psi }_p\right)$ is given by

${\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\mathrm{aAi}\left(-\mathrm{\alpha x}\right)+{\mathrm{b}}^{\text{'}}\mathrm{Bi}\left(-\mathrm{\alpha x}\right)$ ……………………….(6)

(The same solution as the upward-slope case with $\mathrm{\alpha }\to -\mathrm{\alpha }$as role="math" localid="1658393316907" $\mathrm{\alpha }={\left[2{\mathrm{mV}}^{\text{'}}\left(0\right)/{\mathrm{h}}^2\right]}^{1/3}$we also have,

$$\begin{gathered} \mathrm{p}\left(\mathrm{x}\right)=\sqrt{2\mathrm{m}\left(\mathrm{E}-\mathrm{E}+{\mathrm{V}}^{\text{'}}\left(0\right)\mathrm{x}\right)} \\ \mathrm{p}\left(\mathrm{x}\right)={\mathrm{h\alpha }}^{3/2}\sqrt{\mathrm{x}.} \end{gathered}$$

Using region 2 equation to find constant values.

In region 2, we have

$$\begin{gathered} {\int }_0^{\mathrm{x}}\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}\approx {\mathrm{h\alpha }}^{3/2}{\int }_0^{\mathrm{x}}\sqrt{{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}} \\ =\frac{2}{3}\mathrm{h}{\left(\mathrm{\alpha x}\right)}^{3/2} \end{gathered}$$

Therefore the WKB wave function (Equation (5)) can be written as,

$\mathrm{\psi }\left(\mathrm{x}\right)\approx \frac{\mathrm{F}}{\sqrt{\mathrm{h}}{\mathrm{\alpha }}^{3/4}{\mathrm{X}}^{1/4}}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}$

Meanwhile, using the large- z asymptotic form of the Airy function for large negative $\mathrm{z}\left(\mathrm{z}=-\mathrm{\alpha x}\right)$, the patching wave function (Equation (6)) in overlap region 2 becomes

$$\begin{gathered} {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)\approx \frac{{\mathrm{a}}^{\text{'}}}{\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}\sin \left[\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}+\frac{\mathrm{\pi }}{4}\right]+\frac{{\mathrm{b}}^{\text{'}}}{\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}\cos \left[\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}+\frac{\mathrm{\pi }}{4}\right] \\ {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}\left[-{\mathrm{ia}}^{\text{'}}{\mathrm{e}}^{\mathrm{i\pi }/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}+{\mathrm{ia}}^{\text{'}}{\mathrm{e}}^{=-\mathrm{\pi }/4}{\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}\mathrm{\_}}+{\mathrm{be}}^{{}^{\mathrm{i\pi }/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}}+{\mathrm{be}}^{{}^{=-\mathrm{\pi }/4}}{\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}\right] \\ {\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha x}\right)}^{1/4}}\left[{\mathrm{e}}^{\mathrm{i\pi }/4}\left(\mathrm{b}-\mathrm{ia}\right){\mathrm{e}}^{{}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}}+{\mathrm{e}}^{-\mathrm{\pi }/4}{\left(\mathrm{b}\text{'}+\mathrm{ia}\right)}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{\alpha x}\right)}^{3/2}}\right] \end{gathered}$$

Comparing the two solutions, we see that

$\frac{{\mathrm{b}}^{\text{'}}{\mathrm{ia}}^{\text{'}}}{2\sqrt{\mathrm{\pi }}}{\mathrm{e}}^{\mathrm{i\pi }/4}=\frac{\mathrm{F}}{\sqrt{\mathrm{h\alpha }}}\mathrm{And}{\mathrm{b}}^{{\text{'}}^{\text{'}}}+{\mathrm{ia}}^{\text{'}}=0$

which leads to,

${\mathrm{b}}^{\text{'}}={\mathrm{e}}^{\mathrm{i\pi }/4}\sqrt{\frac{\mathrm{\pi }}{\mathrm{h\alpha }}}\mathrm{F}\mathrm{and}{\mathrm{a}}^{\text{'}}={\mathrm{ie}}^{-\mathrm{i\pi }/4}\sqrt{\frac{\mathrm{\pi }}{\mathrm{h\alpha }}}\mathrm{F}$ …………………(7)

Using region 1 equation to find other constant values.

In region 1, we have

${\int }_{\mathrm{x}}^0\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}\approx {\int }_{\mathrm{x}}^0\sqrt{-{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}}=\frac{2}{3}\mathrm{h}{\left(-\mathrm{\alpha x}\right)}^{3/2},$

Therefore the WKB wave function (Equation (5)) can be written as

$\mathrm{\psi }\left(\mathrm{x}\right)\approx \frac{1}{\sqrt{\mathrm{h}}{\mathrm{\alpha }}^{3/4}{\left(-\mathrm{x}\right)}^{1/4}}\left[{\mathrm{Ce}}^{\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}{\mathrm{D}}^{\text{'}}{\mathrm{e}}^{-\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}\right]$

Meanwhile, using the large- z asymptotic form of the Airy function for large positive $\mathrm{Z}\left(\mathrm{Z}=-\mathrm{\alpha x}\right)$, the patching wave function (Equation (6)) in overlap region 1 becomes,

${\mathrm{\psi }}_{\mathrm{p}}\left(\mathrm{x}\right)\approx \frac{\mathrm{a}\text{'}}{2\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha }-\mathrm{x}\right)}^{1/4}}{\mathrm{e}}^{-\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}+\frac{\mathrm{b}\text{'}}{\sqrt{\mathrm{\pi }}{\left(\mathrm{\alpha }-\mathrm{x}\right)}^{1/4}}{\mathrm{e}}^{\frac{2}{3}{\left(-\mathrm{\alpha x}\right)}^{3/2}}$

Comparing the WKB and patching wave functions in overlap region l, we find

$\mathrm{D}\text{'}=\sqrt{\frac{\mathrm{h\alpha }}{4\mathrm{\pi }}\mathrm{a}}\text{'}\mathrm{And}\mathrm{C}\text{'}=\sqrt{\frac{\mathrm{h\alpha }}{\mathrm{\pi }}}\mathrm{b}\text{'}$

or, putting in Equation (7) for a and b :

$\mathrm{D}\text{'}=\frac{\mathrm{i}}{2}{\mathrm{e}}^{-\mathrm{i\pi }/4}\mathrm{F}\mathrm{And}\mathrm{C}\text{'}={\mathrm{e}}^{-\mathrm{i\pi }/4}\mathrm{F}$

But remember that $\mathrm{D}\text{'}={\mathrm{Ce}}^{\mathrm{\gamma }}$and data-custom-editor="chemistry" $\mathrm{C}\text{'}={\mathrm{De}}^{\mathrm{\gamma }}$, then

$\mathrm{C}=\frac{\mathrm{i}}{2}{\mathrm{e}}^{-\mathrm{i\pi }}/4{\mathrm{e}}^{-\mathrm{\gamma }}\mathrm{F}\mathrm{And}\mathrm{D}={\mathrm{e}}^{-\mathrm{i\pi }/4}{\mathrm{e}}^{\mathrm{\gamma }}\mathrm{F}$ ………………(8)

These are the connection formulas, joining the WKB solutions at either side of the turning point $x_2$.

To find the tunneling probability.

Putting Equation (8) forand back into Equation (4) for A , we find

$\mathrm{A}={\mathrm{ie}}^{-\mathrm{i\pi }/2}\left(\frac{{\mathrm{e}}^{-\mathrm{\gamma }}}{4}+{\mathrm{e}}^{\mathrm{\gamma }}\right)$F

It follows that the tunneling probability is

$$\begin{gathered} \mathrm{T}=\frac{{\left|\mathrm{F}\right|}^2}{{\left|\mathrm{A}\right|}^2} \\ \mathrm{T}=\frac{1}{{\left({\mathrm{e}}^{-\mathrm{\gamma }/4+\mathrm{e\gamma }}\right)}^2}. \end{gathered}$$

In case of a broad, higher barrier, $\gamma$is very large.

Hence, the result for the tunneling probability reduces to$\mathrm{T}\approx {\mathrm{e}}^{-2\mathrm{\gamma }}$ .