Editing WKB approximation (section)

==== Second classical turning point ====
For <math>U_1 > 0</math> ie. increasing potential condition or <math>x=x_2
</math> in the given example shown by the figure, we require the exponential function to decay for positive values of x so that wavefunction for it to go to zero. Considering [[Airy function|Airy functions]] to be the required connection formula, we get:<ref name=":3" />

<math display="block">\begin{align}
\operatorname{Ai} (u)\rightarrow \frac{1}{2\sqrt \pi}\frac{1}{\sqrt[4]{u}} e^{-\frac 2 3 u^{\frac 3 2}}  \quad \textrm{where,} \quad u \rightarrow + \infty \\
\operatorname{Ai}(u) \rightarrow \frac{1}{\sqrt \pi}\frac{1}{\sqrt[4]{u}} \cos{\left(\frac 2 3 |u|^{\frac 3 2} - \frac \pi 4\right)}     \quad \textrm{where,} \quad u \rightarrow -\infty\\
\end{align} </math> 

We cannot use Bairy function since it gives growing exponential behaviour for positive x. When compared to WKB solutions and matching their behaviours at <math>\pm \infty </math>, we conclude:

<math>2B=C=N'   </math>,
<math>D=A=0 </math> and <math>\alpha = \frac \pi 4 </math>.  

Thus, letting some normalization constant be <math>N' </math>, the wavefunction is given for increasing potential (with x) as:<ref name=":1" />


<math>\Psi_{\text{WKB}}(x) = \begin{cases}

  \frac{N'}{\sqrt{|p(x)|}} \cos{(\frac 1 \hbar \int_{x}^{x_2} |p(x)| dx - \frac \pi 4)} & \text{if } x_1 < x < x_2 \\    
  \frac{N'}{2\sqrt{|p(x)|}}\exp{(-\frac 1 \hbar \int_{x_2}^{x}  |p(x)| dx )}  & \text{if } x > x_2\\  
 \end{cases}</math>