PX262 - B5 - quantum tunnelling

considering a particle travelling from $x = - \infty$ , with an energy, $E$ , that encounters a potential barrier: $V = {\begin{cases} V_{0} & f o r x \in [0, b] \\ 0 & e l s e w h e r e \end{cases}$
when $E < V_{0}$ , in quantum mechanics, particle may get through
considering the wavefunctions in the regions:

\begin{aligned} ϕ_{I} (x) & = A e^{i k x} + B e^{- i k x} \\ ϕ_{I I} (x) & = C e^{κ x} + D e^{κ x} \\ ϕ_{I I I} (x) & = F e^{i k x} \end{aligned}

where, $κ = \sqrt{\frac{2 m (V_{0} - E)}{ℏ^{2}}}$

\begin{matrix} ϕ_{I} (0) = ϕ_{I I} (0) & ϕ_{I I} (b) = ϕ_{I I I} (b) \\ ϕ_{I}^{'} (a) = ϕ_{I I}^{'} (a) & ϕ_{I I}^{'} (b) = ϕ_{I I I}^{'} (b) \end{matrix}

using these conditions, the probability of the particle passing through the barrier is: $T = \frac{| F |^{2}}{| A |^{2}} = \frac{16 E (V_{0} - E)}{V_{o}^{2}} e^{- 2 κ b}$
phenomena like $α -$ decay, nuclear fission can be explained only using quantum tunnelling