A method for finding tunneling probability for a barrier that is “wide” but whose height v...

A method for finding tunneling probability for a barrier that is “wide” but whose height varies in an arbitrary way is the so-called WKB approximation.

Here U(x) is the height of the arbitrary potential energy barrier, which a particle first penetrates at x = 0 and finally exits at, x = L. Although not entirely rigorous, show that this can be obtained by treating the barrier as a series of rectangular slices, each of width dx (though each is still a “wide” barrier), and by assuming that the probability of tunneling through the total is the product of the probabilities for each slice.