Quantum Tunneling

Quantum tunneling is a phenomenon in quantum mechanics in which a particle or collective quantum degree of freedom passes through a potential-energy barrier despite having insufficient energy to cross it according to classical mechanics. Its mathematical basis is the extension of a quantum state into a classically forbidden region, allowing a nonzero transmission probability through a finite barrier. Tunneling explains phenomena ranging from radioactive decay to electrical transport through thin insulating layers and provides the operating principle of several electronic and microscopic instruments. (openstax.org)

Physical basis

In classical mechanics, a particle with total energy $E$ cannot enter a region where its potential energy $V$ exceeds $E$: its kinetic energy would have to be negative. Quantum mechanics instead describes the particle using a wave function, whose squared magnitude gives its position probability density. This function need not vanish at the boundary of a classically forbidden region. Inside a simple barrier it has exponential, rather than oscillatory, spatial dependence. If the barrier has finite width, a transmitted component can exist beyond it. (openstax.org)

Tunneling differs from thermal activation, in which a system acquires enough energy to pass over a barrier. It is possible even without such an energy increase. In the ideal model of a static barrier and elastic scattering, incident and transmitted particles have the same total energy: the effect follows from a fixed-energy wave equation, not from temporarily acquiring extra energy. (live.ocw.mit.edu)

Mathematical description

For a nonrelativistic particle moving in one dimension, the time-independent Schrödinger equation is

where $m$ is the particle’s mass and $\hbar$ is the reduced Planck constant. For a rectangular barrier of height $V_0>E$ and width $a$, with zero potential on either side, define

Matching the wave function and its derivative at both boundaries gives the transmission coefficient

Here $T$ is the transmitted probability current divided by the incident probability current. (live.ocw.mit.edu)

For an opaque barrier, the dominant dependence is $T\propto e^{-2\kappa a}$. Consequently, increasing barrier width, particle mass, or the energy difference $V_0-E$ strongly suppresses transmission. For a smoothly varying barrier, the WKB approximation gives the leading exponential behavior:

where $x_1$ and $x_2$ are classical turning points satisfying $V(x)=E$. This approximation captures exponential suppression but may require prefactor corrections and becomes unreliable in some regimes, including near the barrier maximum. (ocw.mit.edu)

Historical development

Barrier penetration was analyzed by Friedrich Hund in 1927, shortly after the development of wave mechanics. In 1928, George Gamow, and independently Ronald Gurney and Edward Condon, applied tunneling to alpha decay, explaining how particles escape an atomic nucleus despite the surrounding electrostatic barrier. Their calculations accounted for the strong relationship between emitted-particle energy and radioactive half-life. (openstax.org)

Solid-state tunneling became experimentally established through Leo Esaki’s semiconductor work in the late 1950s and Ivar Giaever’s experiments on superconducting junctions in 1960. Brian Josephson predicted additional superconducting tunneling phenomena in 1962. These contributions received the 1973 Nobel Prize in Physics. (nobelprize.org)

Nuclear phenomena

In alpha decay, a helium nucleus escapes through a barrier produced by the attractive nuclear interaction and electrostatic repulsion. The escaping particle’s energy is below the barrier maximum, but its wave function extends through the forbidden region. Because transmission depends exponentially on barrier properties, relatively modest differences in decay energy can correspond to very large differences in half-life. This is an important application of tunneling in nuclear physics and the explanation of radioactivity. (openstax.org)

Tunneling also enables nuclear fusion below the Coulomb barrier. Colliding positively charged nuclei can penetrate their repulsive barrier and reach separations where the nuclear interaction becomes effective. In heavy-ion fusion, internal nuclear excitations couple to the relative motion, so a single, fixed barrier may be inadequate: multiple coupled reaction channels can substantially alter transmission and fusion probabilities. (arxiv.org)

Electronic devices and microscopy

An electron can tunnel between conducting regions separated by a sufficiently thin barrier. In a semiconductor tunnel diode, this mechanism produces a distinctive current–voltage characteristic that includes a region of negative differential resistance. In superconducting tunnel junctions, measurements of current versus voltage reveal the excitation-energy gap associated with superconductivity, making tunneling a method of spectroscopy. (nobelprize.org)

The scanning tunneling microscope places a sharp conducting tip extremely close to a conducting sample. A voltage drives a tunneling current across the gap. Its strong sensitivity to tip–sample separation allows atomic-scale surface imaging. Developed by Gerd Binnig and Heinrich Rohrer in 1981, the instrument earned them a share of the 1986 physics Nobel Prize. (openstax.org)

Superconducting and collective tunneling

In a tunnel-type Josephson junction, two superconductors are separated by a thin insulating barrier. Coherent transfer of Cooper pairs produces the Josephson effect, including a supercurrent at zero applied voltage and an oscillating current under a constant voltage. (nobelprize.org)

Macroscopic quantum tunneling concerns the escape of a collective variable, such as the superconducting phase difference, from a metastable effective potential well. “Macroscopic” refers to the collective quantum system involved, not to an ordinary large object passing intact through a wall. Experiments with superconducting circuits established tunneling and energy quantization in systems involving very large numbers of Cooper pairs, contributing to the foundations of superconducting qubit technology. (nobelprize.org)