The path integral semi-classical propagator from \(x'\) to \(x\) can be written as \cite{Freed1972}:
\[\begin{split} K(x,t;x',0) = \sum \left[ \frac{1}{2\pi i \hbar} \frac{\partial^2 S(x,t;x',0)}{\partial x \partial x'} \right]^{1/2} \times \\ exp \left[ \frac{1}{i \hbar} S(x,t;x',0) - 1/2 (i \pi) \right] \end{split}\]
Taking the Fourier transform, we obtain the semi-classical Green’s function :
\[\begin{aligned} \begin{split} G(x,x';E) = \\ (i\hbar)^{-1} \int_0^\infty dt. exp \left[ \frac{iEt}{\hbar} \right] K(x,t;x',0) \\ = (i\hbar)^{-1} \sum \left| \frac{\partial^2 W(x,x';E)}{\partial E \partial x'} \frac{\partial^2 W(x,x';E)}{\partial E \partial x} \right]^{1/2} \\ \times exp \left[ \frac{1}{i \hbar} W(x,x';E) - 1/2 (i \pi) \right] \end{split}\end{aligned}\]
where \(W\) is the classical action. Even though this seems counter-intuitive as the motion through barrier is classically forbidden, we can work around this by moving into imaginary time \cite{Freed1972}.
\[W(x,x';E) = \int_{x'}^{x} {2m[E-V(x")]}^{1/2} dx"\]
where \(V(x)\) is the barrier potential.
We now analyze motion of an electron incident on the barrier profile as shown in Fig \ref{fig:paths}. We can look at the green’s function component \(g_i\) of each path \(t_i\) separately, where i denotes the number of reflections within the barrier well.
\[\begin{aligned} & g_0 = \frac{1}{i\hbar} \left[ \frac{m}{2E}\right]^{1/2} exp \left[ \frac{i}{\hbar} p[(x_0-x') + L + (x-x_3)] - 2\kappa l \right] \\ & g_1 = g_0 [1 - exp(-2\kappa l)] exp \left[ \frac{i}{\hbar} \oint p dx" - i\pi \right] \\ & g_n = g_0 [1 - exp(-2\kappa l)]^n exp \left[ n \left[ \frac{i}{\hbar} \oint p dx" - i\pi \right] \right]\end{aligned}\]
where
\[\begin{aligned} p(x) & = (2mE)^{1/2} \\ q(x) & = [2m(V_0-E)]^{1/2} = \hbar \kappa \\ \oint p dx" & = 2(2mE)^{1/2}L\end{aligned}\]
It can be observed that the factor \([1 - exp(-2\kappa l)]\) is the reduced probability amplitude after a set of two reflections at the barriers. This therefore multiplies as we consider paths with more reflections. Summing the contributions from \(t_0\) to \(t_{infty}\), we get the Green’s function as :
\[\begin{split} G(x,x';E) = & \frac{1}{i\hbar} \times \\ & \left[ \frac{m}{2E} \right]^{1/2} exp \left[ \frac{i}{\hbar} p[(x_0-x') + L + (x - x_3)] \right] \times \\ Z(E) \end{split}\]
where,
\[\begin{aligned} Z(E) &= \frac{exp(-2\kappa l)}{1-[1-exp(-2 \kappa l)] exp(i \phi)} \\ \phi &= \frac{1}{\hbar} \oint p dx" - \pi\end{aligned}\]
The transmission coefficient is hence given by :
\[\begin{aligned} T(E) & = |Z(E)|^{2} \\ & = \left| \frac{exp(-2\kappa l)}{1-[1-exp(-2 \kappa l)] exp(i \phi)} \right|^2\end{aligned}\]