Editing Path integral formulation (section)

== Quantum tunneling ==
[[Quantum tunnelling]] can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the [[WKB approximation]], the tunneling rate ({{math|Γ}}) can be determined to be of the form
: <math> \Gamma = A_\mathrm{o} \exp \left(-\frac{S_\mathrm{eff}}{\hbar}\right) </math>
with the effective action {{math|''S''<sub>eff</sub>}} and pre-exponential factor {{math|''A''<sub>o</sub>}}. This form is specifically useful in a [[dissipative system]], in which the systems and surroundings must be modeled together. Using the [[Langevin equation]] to model [[Brownian motion]], the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling.<ref>{{harvnb|Caldeira|Leggett|1983}}</ref> From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted.