= Solution
\b[<Quantum tunnelling> is supported by transport and decay through barriers that classical particles of the same energy cannot traverse.] Three complementary examples are:
* In \b[<alpha decay>], an alpha cluster escapes the attractive nuclear region through a Coulomb barrier. The barrier penetration factor depends exponentially on a forbidden-region action, helping explain the strong dependence of lifetimes on emitted alpha energy. This is evidence from systematic decay rates, rather than direct imaging of an alpha inside the barrier.
* A \b[<scanning tunnelling microscope>] measures current across a narrow <vacuum> gap between a conducting tip and sample. At small bias an approximate dependence is $I\propto e^{-2\kappa s}$ with $\kappa\simeq\sqrt{2m\Phi}/\hbar$. The strong gap-distance sensitivity enables atomic-scale imaging and local spectroscopy. Electronic <density of states> and barrier height also affect the current, so constant-current contrast is not an unqualified geometric atom map.<Surface Studies by Scanning Tunneling Microscopy by Gerd Binnig and others>[Vacuum-tunnelling measurement and surface imaging]{ref}
* A heavily doped \b[<Tunnel diode>] has a narrow p-n barrier. Bias changes the overlap of occupied and empty states across it; interband tunnelling produces a characteristic peak and <Negative differential resistance> interval in its current-voltage curve. The measured voltage dependence agrees with the changing band overlap, providing a solid-state transport example.<New Phenomenon in Narrow Germanium Junctions by Leo Esaki>[Germanium-junction tunnelling characteristic]{ref}
Together these test barrier penetration in nuclear decay, vacuum-gap transport and semiconductor transport. Their detailed currents depend on state supply and matching as well as the barrier transmission probability.
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