Quantum tunnelling Totally Explained

Everything about Quantum Tunnelling totally explained

In quantum mechanics, quantum tunneling is a micro nanoscopic phenomenon in which a particle violates the principles of classical mechanics by penetrating or passing through a potential barrier or impedance higher than the kinetic energy of the particle. A barrier, in terms of quantum tunnelling, may be a form of energy state analogous to a "hill" or incline in classical mechanics, which classically suggests that passage through or over such a barrier would be impossible without sufficient energy.
On the quantum scale, objects exhibit wave-like behaviour; in quantum theory, quanta moving against a potential energy "hill" can be described by their wave-function, which represents the probability amplitude of finding that particle in a certain location at either side of the "hill". If this function describes the particle as being on the other side of the "hill", then there's the probability that it has moved through, rather than over it, and has thus "tunnelled".

History

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunneling. Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. Under this system, it takes an enormous amount of energy to pull apart the nucleus. In quantum mechanics, however, there's a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half-life of the particle and the energy of the emission.
   Alpha decay via tunneling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter, both groups considered whether particles could also tunnel into the nucleus.
   After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunneling. He realized that the tunneling phenomenon wasn't restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Today the theory of tunneling is even applied to the early cosmology of the universe.
   Quantum tunneling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunneling. Tunneling is a source of major current leakage in Very-large-scale integration (VLSI) electronics, and results in the substantial power drain and heating effects that plague high-speed and mobile technology.
   Another major application is in electron-tunneling microscopes (see scanning tunneling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunneling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunneling electrons.
   It has been found that quantum tunneling may be the mechanism used by enzymes to speed up reactions in lifeforms to millions of times their normal speed.

Semi-classical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential

V(x)

. ť

-frac

Where

x_1,x_2

are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck's constant, abbreviated as

hbar 
ightarrow 0

, we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential.

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