where

(338) |

where is a complex constant. Note that this solution represents a particle propagating in the positive -direction [since the full wavefunction is multiplied by , where ] with the continuously varying wavenumber . It follows that

(340) |

where . A comparison of Eqs. (337) and (341) reveals that Eq. (339) represents an approximate solution to Eq. (337) provided that the first term on its right-hand side is negligible compared to the second. This yields the validity criterion , or

In other words, the variation length-scale of , which is approximately the same as the variation length-scale of , must be

According to the WKB solution (339), the probability
density remains constant: *i.e.*,

(343) |

where

(345) |

According to the WKB solution (344), the probability
density *decays exponentially* inside the barrier: *i.e.*,

(346) |

(347) |

We can interpret the ratio of the probability densities to the right and to the left of the potential barrier as the probability, , that a particle
incident from the left will tunnel through the barrier and
emerge on the other side: *i.e.*,

Note that the criterion (342) for the validity of the WKB approximation
implies that the above transmission probability is *very small*. Hence,
the WKB approximation only applies to situations in which there is
very little chance of a particle tunneling through the potential barrier in question.
Unfortunately, the validity criterion (342) breaks down completely
at the edges of the barrier (*i.e.*, at and ), since
at these points. However, it can be demonstrated that the
contribution of those regions, around and , in which the WKB
approximation breaks down to the integral in Eq. (348)
is fairly negligible. Hence, the above expression for the tunneling
probability is a reasonable approximation provided that the incident particle's
de Broglie wavelength is much smaller than the spatial extent of the potential
barrier.