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'''Quantum tunneling''' is ability of a particle to overcome a potential barrier, even though it does not have sufficient [[energy]] to do so. This is conceptually similar to a [[Resurrection]], changing from one physical status to another contrary to traditional laws of [[physics]].

'''Quantum tunneling''' is the ability of a particle to overcome a potential barrier, even though it does not have sufficient [[energy]] to do so. This is conceptually similar to a [[Resurrection]], changing from one physical status to another contrary to traditional laws of [[physics]].

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.

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 tunneling is important in [[radioactivity|radioactive decay]], when an unstable nucleus emits a particle, it utilizes quantum tunneling 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 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 tunneling is forbidden under classical physics. The instantaneous nature of quantum tunneling seems contrary to the [[theory of relativity]], but some argue that it is compatible.<ref>http://m.phys.org/news/2015-05-physicists-quantum-tunneling-mystery.html</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>

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>

This section has advanced mathematical concepts such as [[complex numbers]] and [[hyperbolic trigonometric functions]].

This section has advanced mathematical concepts such as [[complex numbers]] and [[hyperbolic trigonometric functions]].

First, We start from the split wavefunction above, remembering that we have already shown <math>F=0</math>. Then the continuity conditions of a wavefunction (the wavefunction and its first derivative must be continuous) to find four simultaneous equations:

First, we start from the split wavefunction above, remembering that we have already shown <math>F=0</math>. Then the continuity conditions of a wavefunction (the wavefunction and its first derivative must be continuous) result in four simultaneous equations:

<math>\psi_1 (0) = \psi_2 (0)</math><br/>

<math>\psi_1 (0) = \psi_2 (0)</math><br/>

EE^* = \frac{1}{1 - \frac{1}{4} \sin{k_1 a} \frac{k^4_0 -2k^2_0 k^1_0 +k^4_1}{k^2_0 k^2_1}}

EE^* = \frac{1}{1 - \frac{1}{4} \sin{k_1 a} \frac{k^4_0 -2k^2_0 k^1_0 +k^4_1}{k^2_0 k^2_1}}

</math>

</math>

<math>

<math>