Make the step a barrier of finite width and something classically impossible happens: a particle without the energy to climb the barrier sometimes appears on the far side. Inside the barrier the wavefunction decays but does not vanish, and whatever is left at the far wall carries on as a wave. The chance falls off exponentially with the width and with the square root of the energy deficit — so steeply that it decides the rates of nuclear decay and the current in a tunnelling microscope.

Sound gets through a wall not by climbing over it but by being weakened as it passes through. A thicker wall weakens it far more: each extra layer multiplies the loss.

The exact probability density for a particle of energy $E=1$ meeting a barrier of height $2.5$ and width $1.2$, between the dashed lines, in units with $\hbar^{2}/2m=1$. On the left the incoming and reflected waves interfere; inside, the density decays; on the right a steady transmitted wave carries about a fifth of the current on.

For a rectangular barrier of height $V_{0}$ and width $a$ and energy $E<V_{0}$, matching at both edges gives an exact transmission probability. For a thick barrier, , it is dominated by an exponential in the width.

• $\kappa=\sqrt{2m(V_{0}-E)}/\hbar$: a heavier particle or a larger deficit makes larger.

For a thick barrier the hyperbolic sine is dominated by its growing exponential, which makes the second term of the bracket enormous. The inverse of one plus a huge number is its reciprocal, which is the prefactor times $e^{-2\kappa a}$.

For $\kappa a\gg1$ the decaying exponential is negligible.

The second term of the bracket is huge.

One is negligible beside it.

Invert the approximated term.

A particle with less energy than a barrier can still appear on the far side, because its wavefunction decays inside the barrier without vanishing.

For a thick barrier the transmission is dominated by $e^{-2\kappa a}$. Each extra thickness multiplies $T$ by the same factor.

For $\kappa a\gg1$, $T$ is about $16\dfrac{E}{V_{0}}\Bigl(1-\dfrac{E}{V_{0}}\Bigr)e^{-2\kappa a}$.

$\kappa$ grows as the square root of the mass, so a heavier particle meets an exponentially stronger suppression.

Great work! You now know how to:

• explain tunnelling through a barrier
• estimate transmission through a thick barrier
• judge how width, height and mass change the rate