Quantum tunneling

Classically forbidden, quantum allowed

In classical mechanics, a particle with total energy $E$ cannot enter a region where potential energy $V$ is greater than $E$, because kinetic energy would be negative:

$K=E-V<0$

Quantum mechanics changes this conclusion. In a region where $E<V$, the wavefunction does not vanish immediately. Instead, it decays exponentially. If the barrier has finite width, the wavefunction can emerge on the other side with nonzero amplitude. This is quantum tunneling.

Barrier setup

Consider a particle approaching a rectangular barrier:

$V(x)=V_0quad 0<x<a$

and $V=0$ outside. For tunneling, assume

$E<V_0$

Inside the barrier, the time-independent Schrödinger equation gives exponential solutions rather than oscillatory ones. Define

kappa=rac{sqrt{2m(V_0-E)}}{hbar}

The wavefunction inside includes terms like

$e^{-kappa x}$

The decay length is roughly $1/kappa$.

Transmission probability

For a thick or high barrier, the transmission probability is approximately proportional to

$Tsim e^{-2kappa a}$

This shows that tunneling is extremely sensitive to barrier width $a$, particle mass $m$, and the energy gap $V_0-E$.

A slightly wider barrier can reduce tunneling dramatically. Lighter particles tunnel more easily than heavier particles.

Boundary matching

As with the finite square well, the wavefunction and its derivative must be continuous at finite potential boundaries:

$psi_{left}=psi_{barrier}$

rac{dpsi_{left}}{dx}=rac{dpsi_{barrier}}{dx}

and similarly at the second boundary. These conditions determine reflected and transmitted amplitudes.

Not borrowing energy

Tunneling is sometimes described as a particle borrowing energy to cross a barrier, but this is misleading. The stationary-state energy remains $E$. The wavefunction penetrates the classically forbidden region because the Schrödinger equation allows exponential behavior there.

No conservation law is violated.

Alpha decay

Alpha decay in nuclei is a major physical example. An alpha particle is trapped by the nuclear potential but can tunnel through the Coulomb barrier. The tunneling probability determines the decay rate.

Small differences in barrier width and energy can produce enormous differences in half-life because of the exponential dependence.

Scanning tunneling microscope

A scanning tunneling microscope uses electron tunneling between a sharp conducting tip and a surface. The tunneling current depends exponentially on tip-surface distance:

$Ipropto e^{-2kappa d}$

This extreme sensitivity allows imaging of surfaces with atomic-scale resolution.

Other applications

Tunneling appears in nuclear fusion, tunnel diodes, Josephson junctions, field emission, chemical reactions, and semiconductor devices. It is not a rare curiosity; it is central to modern technology and microscopic physics.

The big idea

Quantum tunneling occurs because wavefunctions penetrate finite classically forbidden barriers. The transmission probability decreases roughly as $e^{-2kappa a}$, making tunneling highly sensitive to mass, barrier height, and width. Tunneling explains phenomena from nuclear decay to atomic-resolution microscopy.