The impossible crossing

An energy barrier

Imagine a ball thrown towards a hill. If its kinetic energy is smaller than the energy it would need to reach the top, it climbs a little, slows down, then rolls back down. It does not get to the other side. That is common sense, and it is also the law of classical physics.

   Vue classique :
                          _____
                         /     \\
       bille  --o-->    /       \\
   _______________....-/         \\-....__________
                    (elle monte, ralentit, REDESCEND)

   Energie de la bille < hauteur de la barriere  ->  elle rebrousse chemin

This hill is called a potential barrier: a region requiring an energy that the particle does not have. Classically, a particle without enough energy is stuck on one side. Always. Without exception.

The quantum world says otherwise

At the quantum scale, the particle is not a ball: it is a wave (recall wave-particle duality). And a wave does not stop dead at a wall: it penetrates it a little, gradually weakening. If the barrier is thin enough, the wave does not have time to die out completely: a small part emerges on the other side.

   Vue quantique :
                          _________
        onde incidente   |         |    onde transmise
        \\/\\/\\/\\/\\--->|~~~~~~~~~|--->  /\\/\\  (plus faible,
                         |  \\      |            mais NON NULLE !)
   ______________________|   '-.__ |________________
                         barriere
                         (l'onde s'affaiblit dedans,
                          mais ne s'annule pas tout a fait)

This is quantum tunnelling: the particle has a non-zero probability of ending up on the other side of the barrier, even though it did not have the energy to cross it. As if it had dug a "tunnel" straight through.

This is not energy cheating

Beware a misunderstanding: the particle does not "borrow" energy to jump over. It does not go over at all. It passes through, and comes out with exactly the same energy it had going in. Nothing is violated — not the conservation of energy, nor anything else. It is simply that, for a wave, "being blocked" does not mean "zero probability on the other side".

A question of probability

Tunnelling does not say that a given particle will cross for certain. It says the particle has a certain probability of crossing. Over a very large number of particles, a well-defined fraction gets through, the rest bounce back:

   1000 particules arrivent sur la barriere
        |
        +--> quelques-unes TRAVERSENT (effet tunnel)
        |
        +--> la grande majorite REBONDIT

This probability can be tiny or appreciable, depending on the barrier — that is the subject of the next lesson.

In summary

Classically, a particle without enough energy cannot cross a potential barrier. But because a quantum particle is a wave, it penetrates the barrier while weakening, and if the barrier is thin enough, part of the wave emerges on the other side: this is quantum tunnelling. The particle does not jump over; it passes through, with a certain probability, without violating anything.