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'''Quantum tunnelling''' is ability of a particle to overcome a potential barrier, even though it does not have sufficient [[energy]] to do so. An example is an [[electron]] being fired at one side of a barrier and reappearing on the other side.  The most common example is when the barrier is an insulator, and an electron on one side of the insulator moves to the other side.  The term "tunnelling" is a bit of a misnomer because the electron does not actually travel through the insulator (insulators do not conduct electricity). It reappears with the same energy it started with.

'''Quantum tunneling''' is ability of a particle to overcome a potential barrier, even though it does not have sufficient [[energy]] to do so. An example is an [[electron]] being fired at one side of a barrier and reappearing on the other side.  The most common example is when the barrier is an insulator, and an electron on one side of the insulator moves to the other side.  The term "tunneling" is a bit of a misnomer because the electron does not actually travel through the insulator (insulators do not conduct electricity). It reappears with the same energy it started with.

Quantum tunnelling is important in [[radioactivity|radioactive decay]], when an unstable nucleus emits a particle, it utilizes quantum tunnelling to do so.

Quantum tunneling is important in [[radioactivity|radioactive decay]], when an unstable nucleus emits a particle, it utilizes quantum tunneling to do so.

Quantum tunnelling is forbidden under classical physics. The instantaneous nature of quantum tunnelling appears to defy the [[theory of relativity]], but it is in fact compatible.<ref>http://m.phys.org/news/2015-05-physicists-quantum-tunneling-mystery.html</ref>

Quantum tunneling is forbidden under classical physics. The instantaneous nature of quantum tunneling appears to defy the [[theory of relativity]], but it is in fact compatible.<ref>http://m.phys.org/news/2015-05-physicists-quantum-tunneling-mystery.html</ref>

Quantum tunnelling is considered instanteous in the Copenhagen Interpretion, which is the orthodox view of quantum mechanics. In an experiment published in July 2020, rubidium atoms tunnelled through a 1.3 micrometer barrier in 0.61 milliseconds.<ref>"[https://www.nature.com/articles/s41586-020-2490-7 Measurement of the time spent by a tunnelling atom within the barrier region]," ''Nature'', July 2020.</ref>

Quantum tunneling is considered instantaneous in the [[Copenhagen interpretation]], which is the orthodox view of quantum mechanics. In an experiment published in July 2020, rubidium atoms tunneled through a 1.3 micrometer barrier in 0.61 milliseconds.<ref>"[https://www.nature.com/articles/s41586-020-2490-7 Measurement of the time spent by a tunneling atom within the barrier region]," ''Nature'', July 2020.</ref>

==Mathematics==

==Mathematics==

To see a quantum tunnelling effect, consider a rectangular potential barrier of width <math>a</math>, <math>U(x)</math>, defined as:

To see a quantum tunneling effect, consider a rectangular potential barrier of width <math>a</math>, <math>U(x)</math>, defined as:

<math>

<math>

<math>\psi'_2 (a) = \psi'_3 (a)</math>

<math>\psi'_2 (a) = \psi'_3 (a)</math>

Since there are five variables, only four equations and solutions are linear, we can set <math>A</math> equal to 1. This means that we need to find <math>E</math> to determine the tunnelling probability. This gives the equations (with some rearrangement):

Since there are five variables, only four equations and solutions are linear, we can set <math>A</math> equal to 1. This means that we need to find <math>E</math> to determine the tunneling probability. This gives the equations (with some rearrangement):

<math>1+B=C+D</math><br/>

<math>1+B=C+D</math><br/>