Quantum tunnelling and the transmission probability

Specify the potential before comparing transmission probabilities. Consider a rectangular barrier of width L and height U0, with zero potential on either side and constant particle mass m. Choose an incident energy between zero and U0. Classical passage over the barrier is unavailable, but quantum scattering can produce a transmitted outgoing wave.

Transmission probability T compares transmitted particle flux with incident flux. It is not the wavefunction amplitude itself. Inside this finite barrier, the wavefunction contains growing and decaying exponential terms. Their coefficients follow from matching the wavefunction and its derivative at both edges.

Define κ=√[2m(U0-E)]/ℏ. When κL is much greater than one, transmission has the approximate form C exp(-2κL). The factor C comes from boundary matching. Width appears directly in the exponent; mass and the energy shortfall U0-E enter through a square root.

For width changes with the other parameters fixed, the approximate model gives T(L/2)/T(L)≈exp(κL). This ratio depends on the initial attenuation. Halving the width does not necessarily double transmission or increase it by many orders of magnitude.

An alpha particle contains two protons and two neutrons. Emitting it changes a parent nucleus into a daughter with two fewer protons and two fewer neutrons. In the tunnelling description of alpha decay, the alpha particle escapes a quasibound state through a Coulomb barrier. Its energy can be below the barrier top.

An escape model separates two quantities: the frequency of escape attempts and the probability of transmission per attempt. For small transmission, their product estimates the decay rate. Hold the attempt rate fixed and reduce transmission: the decay rate falls, and the half-life increases. Comparisons among real nuclei also involve changes in alpha energy and barrier shape. Physical barrier width alone does not determine their half-lives.

An STM uses a voltage to drive electron tunnelling between a sharp tip and a conducting sample. The vacuum gap acts as a barrier. Under comparable electronic conditions and fixed bias, increasing the gap reduces current roughly exponentially.

This sensitivity, together with a suitable sharp tip, permits atomic-scale resolution. The current also depends on the electronic states available in the tip and sample. Interpreting an STM image therefore requires more than assigning each contrast change to surface height.

The preceding finite-well model has exterior regions that extend to infinity at potential U0. A bound-state energy lies below U0. Its wavefunction has decaying exterior tails, but its stationary probability density does not drain out of the well. Nonzero density beyond an edge is not evidence of continuing escape.

A finite-width barrier provides a different arrangement. Beyond the forbidden interval, a classically allowed region supports an outgoing transmitted wave. An incoming particle can undergo this scattering process without starting in a bound state. A beam incident on a barrier is one example.

A quasibound state, such as the state used to describe alpha escape, allows decay into an accessible exterior region. An attempt-rate model can estimate this escape, but its repeated attempts should not be treated as a record of measured classical impacts against a wall.

For elastic scattering from a static potential, total energy remains fixed. Tunnelling does not require borrowed energy. The potential may differ between the incoming and outgoing regions, so conserved total energy does not by itself guarantee equal kinetic energies. Those kinetic energies are equal when the exterior potentials are equal.