Quantum tunnelling is a quantum mechanical phenomenon in which a particle has a nonzero probability of being detected in a region that would be classically inaccessible because the particle lacks sufficient energy to cross a potential barrier. In classical mechanics, a particle with energy lower than a barrier height cannot pass over the barrier if no other mechanism supplies additional energy.
Quantum mechanics instead describes the particle using a wave function or quantum state that can extend into and sometimes through the classically forbidden region. The amplitude generally decreases inside the barrier, but for a barrier of finite width it can remain nonzero on the far side. Consequently, there is a finite probability of transmission. Tunnelling does not mean that a particle drills a literal hole through a barrier. It is a consequence of the mathematical behaviour of quantum states in spatially varying potentials.
The simplest tunnelling model involves a particle encountering a rectangular potential barrier of height V0 greater than the particle energy E and width a. Inside the barrier, the time independent Schrodinger equation produces an exponentially varying wave function rather than an ordinary oscillating wave. The decay constant depends on the square root of the difference between barrier height and particle energy, divided by hbar and the particle mass.
The transmission probability decreases approximately exponentially as the barrier becomes wider or higher. A commonly encountered qualitative form is T proportional to exp of negative 2 kappa a, where kappa depends on the particle mass, barrier height and particle energy. This exponential dependence explains why tunnelling can be extremely sensitive to barrier dimensions. Small changes in distance or barrier properties can produce large changes in transmission probability.
Quantum tunnelling occurs in many natural physical systems. Alpha decay is a classic example. An alpha particle inside a radioactive nucleus can be trapped by the nuclear potential and the electrostatic repulsion that forms a barrier. Although the particle may have insufficient classical energy to escape directly over the barrier, its quantum state has a finite tunnelling probability through the barrier.
The rate of alpha decay depends strongly on barrier characteristics, helping explain why different radioactive isotopes can have dramatically different half lives. Tunnelling also contributes to nuclear fusion in stars because positively charged nuclei repel one another electrically. At stellar temperatures, some nuclei can approach closely enough that quantum tunnelling increases the probability of overcoming the effective Coulomb barrier and entering the range where the strong nuclear interaction can act.
Tunnelling is important in semiconductor devices. Tunnel diodes exploit quantum transmission through thin potential barriers and can exhibit electrical characteristics useful in high speed or specialised circuits. Flash memory uses quantum tunnelling during certain programming and erasing operations, where electrons pass through thin insulating barriers under applied electric fields. Scanning tunnelling microscopes exploit tunnelling current between a sharp conducting tip and a surface separated by a very small gap.
The tunnelling current changes extremely rapidly with tip to surface distance, allowing the instrument to map surfaces at atomic scales under suitable conditions. These applications demonstrate that tunnelling is not merely a theoretical effect. Engineers can control barrier widths, voltages and material properties so that tunnelling becomes a useful electrical signal.
Quantum tunnelling also occurs in chemistry and biology. Proton and electron transfer can involve tunnelling contributions when reaction barriers and molecular structures create suitable conditions. In some chemical reactions, tunnelling can alter reaction rates substantially compared with predictions based solely on classical thermal activation. Hydrogen is particularly relevant because its low mass increases the importance of tunnelling relative to heavier nuclei under comparable conditions.
Certain biological processes have been investigated for possible quantum tunnelling contributions, although claims about specific biological mechanisms must be evaluated according to the experimental evidence and the timescales involved. The general principle is well established: quantum particles can have transmission probabilities through barriers that would be forbidden by classical energy considerations.
Tunnelling is closely related to the wave nature of quantum states. The wave function is not a material wave physically leaking through a wall in the classical sense. It is a mathematical amplitude used to calculate probabilities for measurement outcomes. When the potential changes, the wave function must satisfy the quantum mechanical boundary conditions across the interface. The resulting solution can extend into a classically forbidden region and decay exponentially.
If the barrier is finite, the wave function can emerge on the other side with a nonzero amplitude. The square of the appropriate amplitude determines transmission probability. This makes tunnelling a direct consequence of solving the quantum dynamics rather than a separate force that pushes particles through barriers.
The probability of tunnelling depends on mass, energy and barrier properties. Lighter particles generally have larger tunnelling probabilities for comparable barriers because the relevant exponential decay constant increases with the square root of mass. Higher particle energy reduces the effective barrier and increases transmission. A thinner barrier increases transmission dramatically, while a thicker barrier can suppress it exponentially. The detailed transmission coefficient also depends on the shape of the potential rather than only its maximum height. Realistic systems can contain asymmetric barriers, multiple barriers, resonances and time dependent potentials. Resonant tunnelling can occur when intermediate quantum states allow transmission to become especially large at particular energies. These effects are important in nanotechnology and semiconductor physics.
Quantum tunnelling does not violate conservation of energy. The common popular explanation that a particle temporarily borrows energy from the vacuum to cross a barrier is not an accurate general description of ordinary tunnelling. In a stationary tunnelling problem, the particle energy remains a well defined conserved quantity when the potential is time independent, even though the particle has a nonzero probability amplitude inside a region where classical kinetic energy would be negative.
The quantity called kinetic energy in the classical interpretation is not directly transferable to the quantum wave function in the same way. The quantum solution permits exponentially decaying states in the forbidden region without requiring an energy violation. Tunnelling therefore demonstrates the limits of classical trajectory intuition while remaining fully consistent with the conservation laws of quantum theory.
