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Your Universe of Digital Possibilities
A quantum isn’t a pebble — it’s a wave of where it might be. Fire it at a barrier taller than its energy and classical physics says it always turns back. Watch instead: the packet hits the wall, most of it reflects — but inside the barrier the wave doesn’t stop, it decays, and if the wall is thin enough a sliver leaks out the far side and sails on. The particle is simply found, sometimes, where it could never have walked. Raise the wall and the leak dies away.
The wave function’s whole future, set by its curvature and the potential V. We solve it exactly in time by splitting it into a potential kick and a kinetic drift.
Inside a wall too tall (E < V₀) the wave doesn’t oscillate — it decays. Not to zero, just smaller: thin enough, and something survives to the far side.
The exact leak for one sharp energy. The sinh makes it fall exponentially with width a and with √(V₀−E) — why a hair more wall kills it.
For a fat or soft wall, the leak is set by the area of forbidden region the wave must cross — Gamow’s factor, the same integral that dates a rock by its alpha decay.
Probability is conserved: what doesn’t get through comes back. The instrument measures T by integrating |ψ|² past the wall — its own answer for a packet of many energies.
A wall, it turns out, is never a flat no — only a price, and quantum mechanics is always willing to pay it some of the time. This one effect is why the Sun shines (protons fuse only by tunnelling through their own electric repulsion), why a lump of uranium decays on a clock you can read in rock, how a scanning tunnelling microscope feels single atoms, and how the charge leaks into every flash-memory cell you own. The barrier doesn’t forbid — it just makes the far side improbable. And improbable, run often enough, is the same as inevitable.