Editing Quantum tunnelling (section)

=== Tunnelling problem ===
[[File:E14-V20-B1.gif|thumb|A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.|alt=]]

The [[wave function]] of a [[physical system]] of particles specifies everything that can be known about the system.<ref>{{Cite book |last1=Bjorken |first1=James D. |title=Relativistic quantum mechanics |last2=Drell |first2=Sidney D. |date=1964 |publisher=McGraw Hill |isbn=978-0-07-005493-6 |series=International series in pure and applied physics |location=New York, NY |pages=2}}</ref> Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the [[Schrödinger equation]], the time evolution of a known wave function can be deduced. The square of the [[absolute value#Complex numbers|absolute value]] of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.

As shown in the animation, when a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more de-localized: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is ''somewhere'' remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.

Some models of a tunneling barrier, such as the [[rectangular potential barrier|rectangular barrier]]s shown, can be analysed and solved algebraically.<ref name=Messiah>{{Cite book |last=Messiah |first=Albert |url=https://archive.org/details/quantummechanics0000mess/quantummechanics0000mess |title=Quantum Mechanics |date=1966 |publisher=North Holland, John Wiley & Sons |isbn=0486409244 |language=en}}</ref>{{rp|96}} Most problems do not have an algebraic solution, so numerical solutions are used. "[[Semiclassical physics|Semiclassical methods]]" offer approximate solutions that are easier to compute, such as the [[WKB approximation]].