Thick-barrier transmission prefactor ($T\simeq 16E(U-E)U^{-2}{e}^{-2\kappa a}$)

For a thick rectangular barrier with equal exterior potentials, constant mass and $0<E<U$, the exact matched transmission probability has the displayed limit when $\kappa a\gg 1$. The exponential is fixed by the evanescent action; the interface prefactor remains energy-dependent. A leading-exponential estimate can therefore differ by an order-one multiplier from the matched result.