Quantum tunnelling

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${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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v t e

In physics, quantum tunnelling, barrier penetration, or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics, should not be passable due to the object not having sufficient energy to pass or surmount the barrier.

Tunneling is a consequence of the wave nature of matter, where the quantum wave function describes the state of a particle or other physical system, and wave equations such as the Schrödinger equation describe their behavior. The probability of transmission of a wave packet through a barrier decreases exponentially with the barrier height, the barrier width, and the tunneling particle's mass, so tunneling is seen most prominently in low-mass particles such as electrons or protons tunneling through microscopically narrow barriers. Tunneling is readily detectable with barriers of thickness about 1–3 nm or smaller for electrons, and about 0.1 nm or smaller for heavier particles such as protons or hydrogen atoms.^[1] Some sources describe the mere penetration of a wave function into the barrier, without transmission on the other side, as a tunneling effect, such as in tunneling into the walls of a finite potential well.^[2][3]

Tunneling plays an essential role in physical phenomena such as nuclear fusion^[4] and alpha radioactive decay of atomic nuclei. Tunneling applications include the tunnel diode,^[5] quantum computing, flash memory, and the scanning tunneling microscope. Tunneling limits the minimum size of devices used in microelectronics because electrons tunnel readily through insulating layers and transistors that are thinner than about 1 nm.^[6]

The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century.^[7]