Transmission coefficient

Transmission coefficient.

A particle moving along the x-axis, with mass m and energy E, approaches from the left a potential barrier with energy U > E. According to quantum mechanics, the probability that the particle penetrates the potential barrier (blue line in the draw) is not null. The transmission coefficient T is the probability that the particle reaches the point a , thus passing the potential barrier and penetrating into the region on the right.

Let φ (x) = e^{-x k} a solution of the first Schrödinger equation applied to the particle inside the potential barrier, where . A rough approximation for the transmission coefficient is given by the quotient of φ (a) by φ (0). This quotient is e^{-a k} ; the square of its module gives the probability that the particle passes the potential barrier. This probability is, by definition, the transmission coefficient. Thus T = | φ (a) / φ (0) | ² = | e ^{- a k} |² = e ^{- 2 a k} .

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