WKB Approximation and Connection Formula

The WKB approximation, which is sometimes called the JWKB approximation, paying respect to the contribution of mathematician H. Jeffreys, provides us with the analytical expression of wave function for a particle moving in a general potential y(jc) as 

Since this wave function becomes the quantum mechanically exact solution—namely, plane wave, when the potential y(jc) does not depend on the coordinate jc and is constant—the WKB approximation holds well when the potential weakly depends on the coordinate. That is to say, the condition that the above WKB approximation holds well is given by 

At the turning point x = b, the following two connection formulas hold  

Here we can apply the connection formula Equation (2.50) to the last exponential function in the above expression and we have the wave function in the classically allowed region at x as 

The square of the first term of this expression gives the tunneling probability as 

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