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[4. Applications]
Transmission coefficient.
A particle moving along the xaxis, 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} .
