Transmission coefficient (physics)

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For other uses, see Transmission coefficient.

In non-relativistic quantum mechanics, the transmission coefficient and related reflection coefficient are used to describe the behavior of waves incident on a barrier. The transmission coefficient represents the probability flux of the transmitted wave relative to that of the incident wave. It is often used to describe the probability of a particle tunneling through a barrier.

The transmission coefficient is defined in terms of the incident and transmitted probability current density j according to:

$T = \frac{|j_{transmitted}|}{|j_{incident}|},$

where j_incident is the probability current in the wave incident upon the barrier and j_transmitted is the probability current in the wave moving away from the barrier on the other side.

The reflection coefficient R is defined analogously, as R=|j_reflected|/|j_incident|. Conservation of probability implies that T+R=1.

For sample calculations, see finite potential barrier or square potential.

[edit] WKB approximation

Main article: WKB approximation

Using the WKB approximation, one can obtain a tunelling coefficient that looks like

$T = \frac{e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{ \left( 1 + \frac{1}{4} e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} \right)^2}$

Where $x 1, x 2$ are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck's constant, abbreviated as $\hbar \rightarrow 0$ , we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential.