WKB tunnelling exponent ($T\simeq e^{-2S_B/\hslash }$)

For an opaque barrier between simple turning points, the leading transmission probability is $T\simeq e^{-2S_B/\hslash }$, where $S_B=\int \sqrt{2m[V(x)-E]}\mathrm{d}x$ over the forbidden interval. The exponent for probability is twice that for amplitude. Matching determines prefactors and the domain where this approximation applies.