= Solution
A <scanning tunneling microscope> brings a tip and surface, both <electrical conductors>, close enough to leave only a small <vacuum> gap. An applied <electric potential difference> makes occupied <Electron> states on one side able to feed unoccupied states on the other. Although the gap is a classically forbidden <potential barrier>, <quantum tunnelling> through its finite width produces an <electric current>.
For an approximate <rectangular potential barrier> of gap width $d$, the <transmission probability>, and thus the <electric current> at fixed small bias, varies approximately as $e^{-2\kappa d}$. This strong distance dependence makes the device sensitive to changes in separation on the scale of a <chemical atom>. <Feedback control> moves the tip vertically to keep the <electric current> constant during a scan; the required height changes form an image of the surface, with contrast also depending on its electronic states. The finite gap permits transmitted <probability current>, whereas the semi-infinite step in part (b) does not.
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