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The potential barrier extends from$x=0$ to $x=L$.

The height of the potential barrier is $U\left(x\right)$.

In quantum mechanics, there is a possibility that a particle can cross the potential barrier higher than its kinetic energy, by penetrating through it. This phenomenon is known as Tunneling.

Let us consider T= transmission coefficient of a wide potential barrier.

If E is the energy of incident particles and $U_o$is the height of potential barrier, then transmission coefficient can be written as

$T=16\frac{E}{U_o}(1鈭�\frac{E}{U_o})e^{鈭�2L\sqrt{2m(U_o鈭�E)}/\mathrm{鈩�}}$

The total transmission probability for entire potential barrier,considering it to be a series of very thin rectangular slices of width $螖x$, is the product of probability of transmission for each slice.

$\begin{array}{c}{T}_{\text{total}}鈮�\underset{n=0}{\overset{L}{鈭�}}\exp \left[\frac{鈭�2\left(螖x\right)脳\sqrt{2m(U_{掳}鈭�E)}}{\mathrm{鈩�}}\right]\\ {}^{鈮�\exp \left[\underset{n=0}{\overset{L}{鈭�}}\frac{鈭�2\left(螖x\right)脳\sqrt{2m(U_{掳}鈭�E)}}{\mathrm{鈩�}}\right]}\end{array}$

Here we are using Riemann sum method to convert the above equation back to integration with limit x=0 to x=L and replacing potential steps $\left(U_n\right)$with $U_x$(a continuous functions of X)

$T_{\text{total}}=\exp \left[{鈭�}_0^L\frac{鈭�2\sqrt{2m\left(U\left(x\right)鈭�E\right)}}{\mathrm{鈩�}}dx\right]$

The final tunneling probability is$T_{\text{total}}=\exp \left[{鈭�}_0^L\frac{鈭�2\sqrt{2m\left(U\left(x\right)鈭�E\right)}}{\mathrm{鈩�}}dx\right]$