Crossing a forbidden barrier

What governs tunnelling

Three decisive factors

The probability that a particle crosses by tunnelling depends on three things. Understanding which ones, and how, explains why tunnelling is decisive for an electron but completely negligible for a ball.

   Facteur                       Effet sur la probabilite de passage
   ----------------------------  -----------------------------------
   largeur de la barriere (L)    plus large  -> proba S'EFFONDRE
   hauteur de la barriere        plus haute  -> proba diminue
   masse de la particule (m)     plus lourde -> proba S'EFFONDRE

The exponential dependence: the key point

The most important factor is the width of the barrier, and the dependence is not gentle: it is exponential. The probability of crossing decreases as:

   P ∝ e^(-k L)     (k dependant de la masse et de la hauteur)

An exponential decay is brutal. Doubling the width does not divide the probability by two: it can divide it by thousands or millions.

   onde dans la barriere (amplitude qui chute exponentiellement) :

   amplitude
      ^
      |*
      | *
      |  *
      |   **
      |     ***
      |        ****
      |            ******
      |                  *********______
      +-------------------------------------> profondeur dans la barriere
      entree                              sortie

   barriere fine  : l'onde ressort encore appreciable  -> passage possible
   barriere large : l'onde est deja quasi eteinte       -> passage quasi nul

Why a ball never passes through a wall

Apply these rules to a macroscopic object. A ball is enormous in mass compared to an electron, and a wall is thick on the atomic scale. The two factors that collapse the probability (large mass, large width) are maximal:

   electron, barriere de ~1 nm  ->  P notable (l'effet tunnel opere)
   bille, mur de quelques cm    ->  P ~ e^(-nombre gigantesque) ≈ 0

   (attendre le passage d'une bille : bien plus que l'age de l'univers)

The probability of a ball passing through a wall is not strictly zero... but it is so small that you would have to wait infinitely longer than the age of the universe to see it happen once. Tunnelling is real at every scale, but it is only observable at the atomic scale.

The rule to remember

   particule LEGERE + barriere FINE + barriere BASSE  ->  effet tunnel notable
   particule LOURDE + barriere LARGE + barriere HAUTE  ->  effet tunnel nul

It is because the electron is ultra-light and atomic distances are minuscule (the nanometre) that tunnelling becomes a common and exploitable phenomenon in this world — as we will see with its applications.

In summary