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.

Factor	Effect on the probability of getting through
barrier width (L)	wider -> probability COLLAPSES
barrier height	higher -> probability decreases
particle mass (m)	heavier -> probability COLLAPSES

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 depending on the mass and the height)

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

   wave inside the barrier (amplitude dropping exponentially):

   amplitude
      ^
      |*
      | *
      |  *
      |   **
      |     ***
      |        ****
      |            ******
      |                  *********______
      +-------------------------------------> depth into the barrier
      entry                               exit

   thin barrier : the wave still comes out noticeably  -> passage possible
   wide barrier : the wave is already almost extinct   -> passage almost nil

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, barrier of ~1 nm   ->  P significant (tunnelling occurs)
   marble, wall a few cm thick  ->  P ~ e^(-gigantic number) ≈ 0

   (waiting for a marble to get through: far longer than the age of the universe)

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

   LIGHT particle + THIN barrier + LOW barrier    ->  significant tunnelling
   HEAVY particle + WIDE barrier + HIGH barrier   ->  no tunnelling

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