91影视

We use a patching wave function $\left({\mathrm{蠄}}_{\mathrm{p}}\right)$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" $蠄\left(x\right)\left\{\begin{array}{l}\begin{array}{l}\frac{1}{\sqrt{\mathrm{p}\left(\mathrm{x}\right)}}[{\mathrm{Ae}}^{-\frac{\mathrm{i}}{\mathrm{h}}{鈭�}_{\mathrm{x}}^0\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}+{\mathrm{Be}}^{\frac{\mathrm{i}}{\mathrm{h}}{鈭�}_{\mathrm{x}}^0\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}}],\mathrm{x}<0,\\ \frac{1}{\sqrt{\left|\mathrm{p}\left(\mathrm{x}\right)\right|}}[{\mathrm{Ce}}^{\frac{\mathrm{i}}{\mathrm{h}}{鈭�}_0^{\mathrm{x}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}+{\mathrm{De}}^{-\frac{1}{\mathrm{h}}{鈭�}_0^{\mathrm{X}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}}],\mathrm{x}>0.\end{array}\\ \end{array}\right.鈥�鈥�鈥�鈥�鈥�.\left(1\right)$

and the patching wave function is

${\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\mathrm{aAi}\left(\mathrm{伪虫}\right)+\mathrm{bBi}\left(\mathrm{伪虫}\right),鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�..\left(2\right)$

For appropriate constants a and b .

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

$\mathrm{p}\left(\mathrm{x}\right)鈮�\sqrt{2\mathrm{m}\left(\mathrm{E}-\mathrm{E}-{\mathrm{V}}^{,}\left(0\right)\mathrm{x}\right)}={\mathrm{丑伪}}^{3/2}\sqrt{-\mathrm{x}}$

In particular, in overlap region 2,

${鈭�}_0^{\mathrm{x}}\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}鈮�{\mathrm{丑伪}}^{3/2}{鈭�}_0^{\mathrm{x}}\sqrt{{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}}=\frac{2}{3}\mathrm{h}{\left(\mathrm{伪齿}\right)}^{3/2}$

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

$\mathrm{蠄}\left(\mathrm{x}\right)鈮�\frac{1}{\sqrt{\mathrm{h}}{\mathrm{伪}}^{3/4}{\mathrm{蠂}}^{1/4}}\left[{\mathrm{Ce}}^{\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}+{\mathrm{De}}^{-\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}\right]$

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

${\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)鈮�\frac{\mathrm{a}}{2\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}{\mathrm{e}}^{-\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}+\frac{\mathrm{b}}{\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}{\mathrm{e}}^{\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}$

Comparing the two solutions, we see that

$\mathrm{a}=\sqrt{\frac{4\mathrm{蟺}}{\mathrm{丑伪}}}\mathrm{D},\mathrm{And}\mathrm{b}=\sqrt{\frac{\mathrm{蟺}}{\mathrm{丑伪}}}\mathrm{C}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�(3)

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

${\text{鈭�}}_x^{0}p\left(x^{\text{'}}\right)d{x}^{\text{'}}鈮�h{伪}^{3/2}{鈭�}_x^0\sqrt{-x^{\text{'}}}d{x}^{\text{'}}=\frac{2}{3}h{\left(-伪x\right)}^{3/2}$

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

$\mathrm{蠄}\left(\mathrm{x}\right)鈮�\frac{1}{\sqrt{\mathrm{h}}{\mathrm{伪}}^{3/4}{\left(-\mathrm{x}\right)}^{1/4}}\left[{\mathrm{Ae}}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}+{\mathrm{Be}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}\right]$

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

$$\begin{gathered} {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)鈮�\frac{\mathrm{a}}{\sqrt{\mathrm{蟺}}{\left(-\mathrm{伪虫}\right)}^{1/4}}\sin \left[\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}+\frac{\mathrm{蟺}}{4}\right]+\frac{\mathrm{b}}{\sqrt{\mathrm{蟺}}{\left(-\mathrm{伪虫}\right)}^{1/4}}\cos \left[\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}+\frac{\mathrm{蟺}}{4}\right] \\ {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{蟺}}{\left(-\mathrm{伪虫}\right)}^{1/4}}\left[-{\mathrm{iae}}^{\mathrm{c}}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}+-{\mathrm{iae}}^{-\mathrm{颈蟺}//4}{\mathrm{e}}^{-\mathrm{c}}+{\mathrm{be}}^{\mathrm{颈蟺}/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}+{\mathrm{be}}^{-\mathrm{颈蟺}//4}{\mathrm{e}}^{{}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}}\right] \\ {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{蟺}}{\left(-\mathrm{伪虫}\right)}^{1/4}}\left[{\mathrm{e}}^{\mathrm{颈蟺}//4}\left(\mathrm{b}-\mathrm{ia}\right){\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}+{\mathrm{e}}^{\mathrm{颈蟺}//4}\left(\mathrm{b}+\mathrm{ia}\right){\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}\right] \end{gathered}$$

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

We find

$\frac{\mathrm{b}-\mathrm{ia}}{2\sqrt{\mathrm{蟺}}}{\mathrm{e}}^{\mathrm{颈蟺}/4}=\frac{\mathrm{B}}{\sqrt{\mathrm{丑伪}}}\mathrm{and}\frac{\mathrm{b}+\mathrm{ia}}{2\sqrt{\mathrm{蟺}}}{\mathrm{e}}^{-\mathrm{颈蟺}/4}=\frac{\mathrm{A}}{\sqrt{\mathrm{丑伪}}}$

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

$\mathrm{B}=\left(\frac{\mathrm{C}}{2}-\mathrm{iD}\right){\mathrm{e}}^{\mathrm{颈蟺}/4}\mathrm{and}\mathrm{A}=\left(\frac{\mathrm{C}}{2}+\mathrm{iD}\right){\mathrm{e}}^{-\mathrm{颈蟺}/4}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�(4)

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

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 $\mathrm{x}1<\mathrm{x}<\mathrm{x}2$:

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

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

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

In this case, the linear approximation of the potential is

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

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

${\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\mathrm{aAi}\left(-\mathrm{伪虫}\right)+{\mathrm{b}}^{\text{'}}\mathrm{Bi}\left(-\mathrm{伪虫}\right)$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(6)

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

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

In region 2, we have

$$\begin{gathered} {鈭�}_0^{\mathrm{x}}\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right){\mathrm{dx}}^{\text{'}}鈮�{\mathrm{丑伪}}^{3/2}{鈭�}_0^{\mathrm{x}}\sqrt{{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}} \\ =\frac{2}{3}\mathrm{h}{\left(\mathrm{伪虫}\right)}^{3/2} \end{gathered}$$

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

$\mathrm{蠄}\left(\mathrm{x}\right)鈮�\frac{\mathrm{F}}{\sqrt{\mathrm{h}}{\mathrm{伪}}^{3/4}{\mathrm{X}}^{1/4}}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}$

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

$$\begin{gathered} {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)鈮�\frac{{\mathrm{a}}^{\text{'}}}{\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}\sin \left[\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}+\frac{\mathrm{蟺}}{4}\right]+\frac{{\mathrm{b}}^{\text{'}}}{\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}\cos \left[\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}+\frac{\mathrm{蟺}}{4}\right] \\ {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}\left[-{\mathrm{ia}}^{\text{'}}{\mathrm{e}}^{\mathrm{颈蟺}/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}+{\mathrm{ia}}^{\text{'}}{\mathrm{e}}^{=-\mathrm{蟺}/4}{\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}\mathrm{\_}}+{\mathrm{be}}^{{}^{\mathrm{颈蟺}/4}{\mathrm{e}}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}}+{\mathrm{be}}^{{}^{=-\mathrm{蟺}/4}}{\mathrm{e}}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}\right] \\ {\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{1}{2\sqrt{\mathrm{蟺}}{\left(\mathrm{伪虫}\right)}^{1/4}}\left[{\mathrm{e}}^{\mathrm{颈蟺}/4}\left(\mathrm{b}-\mathrm{ia}\right){\mathrm{e}}^{{}^{\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}}+{\mathrm{e}}^{-\mathrm{蟺}/4}{\left(\mathrm{b}\text{'}+\mathrm{ia}\right)}^{-\mathrm{i}\frac{2}{3}{\left(\mathrm{伪虫}\right)}^{3/2}}\right] \end{gathered}$$

Comparing the two solutions, we see that

$\frac{{\mathrm{b}}^{\text{'}}{\mathrm{ia}}^{\text{'}}}{2\sqrt{\mathrm{蟺}}}{\mathrm{e}}^{\mathrm{颈蟺}/4}=\frac{\mathrm{F}}{\sqrt{\mathrm{丑伪}}}\mathrm{And}{\mathrm{b}}^{{\text{'}}^{\text{'}}}+{\mathrm{ia}}^{\text{'}}=0$

which leads to,

${\mathrm{b}}^{\text{'}}={\mathrm{e}}^{\mathrm{颈蟺}/4}\sqrt{\frac{\mathrm{蟺}}{\mathrm{丑伪}}}\mathrm{F}\mathrm{and}{\mathrm{a}}^{\text{'}}={\mathrm{ie}}^{-\mathrm{颈蟺}/4}\sqrt{\frac{\mathrm{蟺}}{\mathrm{丑伪}}}\mathrm{F}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌�(7)

In region 1, we have

${鈭�}_{\mathrm{x}}^0\left|\mathrm{p}\left({\mathrm{x}}^{\text{'}}\right)\right|{\mathrm{dx}}^{\text{'}}鈮�{鈭�}_{\mathrm{x}}^0\sqrt{-{\mathrm{x}}^{\text{'}}}{\mathrm{dx}}^{\text{'}}=\frac{2}{3}\mathrm{h}{\left(-\mathrm{伪虫}\right)}^{3/2},$

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

$\mathrm{蠄}\left(\mathrm{x}\right)鈮�\frac{1}{\sqrt{\mathrm{h}}{\mathrm{伪}}^{3/4}{\left(-\mathrm{x}\right)}^{1/4}}\left[{\mathrm{Ce}}^{\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}{\mathrm{D}}^{\text{'}}{\mathrm{e}}^{-\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}\right]$

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

${\mathrm{蠄}}_{\mathrm{p}}\left(\mathrm{x}\right)鈮�\frac{\mathrm{a}\text{'}}{2\sqrt{\mathrm{蟺}}{\left(\mathrm{伪}-\mathrm{x}\right)}^{1/4}}{\mathrm{e}}^{-\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}+\frac{\mathrm{b}\text{'}}{\sqrt{\mathrm{蟺}}{\left(\mathrm{伪}-\mathrm{x}\right)}^{1/4}}{\mathrm{e}}^{\frac{2}{3}{\left(-\mathrm{伪虫}\right)}^{3/2}}$

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

$\mathrm{D}\text{'}=\sqrt{\frac{\mathrm{丑伪}}{4\mathrm{蟺}}\mathrm{a}}\text{'}\mathrm{And}\mathrm{C}\text{'}=\sqrt{\frac{\mathrm{丑伪}}{\mathrm{蟺}}}\mathrm{b}\text{'}$

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

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

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

$\mathrm{C}=\frac{\mathrm{i}}{2}{\mathrm{e}}^{-\mathrm{颈蟺}}/4{\mathrm{e}}^{-\mathrm{纬}}\mathrm{F}\mathrm{And}\mathrm{D}={\mathrm{e}}^{-\mathrm{颈蟺}/4}{\mathrm{e}}^{\mathrm{纬}}\mathrm{F}$ 鈥︹赌︹赌︹赌︹赌︹赌�(8)

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

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

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

It follows that the tunneling probability is

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

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

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