For tunneling problems in WKB, you have 3 regions I (allowed), II (barrier), III(allowed)

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Rewriting in terms of cosines and sines, you get:

so

Now match region 2 and region 3.

Define . Then

Matching, you get: and

This gives an equation:

and a similar equation

This can be used to get

This means T= for a tall, thick barrier. This means

example:

parabolic potential tunnelling

WKB tunnelling predicts transmission of 1 when E V. This is the classical limit.

source psfile jl@crush.caltech.edu index parabolic_potential_tunneling