Editing WKB approximation (section)

==== First classical turning point ====
For <math>U_1 < 0</math> ie. decreasing potential condition or <math>x=x_1 
</math> in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero. Considering Bairy functions to be the required connection formula, we get:<ref name=":3">{{Cite journal |last1=Ramkarthik |first1=M. S. |last2=Pereira |first2=Elizabeth Louis |date=2021-06-01 |title=Airy Functions Demystified — II |url=https://doi.org/10.1007/s12045-021-1179-z |journal=Resonance |language=en |volume=26 |issue=6 |pages=757–789 |doi=10.1007/s12045-021-1179-z |issn=0973-712X|url-access=subscription }}</ref>

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

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

<math>A=-D=N </math>,
<math>B=C=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)|}}\exp{(-\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx )}  & \text{if }  x < x_1\\ 
  \frac{N}{\sqrt{|p(x)|}} \sin{(\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx - \frac \pi 4)} & \text{if } x_2 > x > x_1 \\
 \end{cases}    </math>