Editing WKB approximation (section)

===Connection conditions===
It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of {{math|''E''}}, this matching procedure will not work: The function obtained by connecting the solution near <math>+\infty</math> to the classically allowed region will not agree with the function obtained by connecting the solution near <math>-\infty</math> to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy {{math|''E''}}, which will give an approximation to the exact quantum energy levels.[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the energy level and its intersection with potential shows the turning points with dotted lines. The problem has two classical turning points with <math>U_1 < 0</math> at <math>x=x_1 
</math> and <math>U_1 > 0</math> at <math>x=x_2
</math>.]]The wavefunction's coefficients can be calculated for a simple problem shown in the figure. Let the first turning point, where the potential is decreasing over x, occur at <math>x=x_1 
</math> and the second turning point, where potential is increasing over x, occur at <math>x=x_2
</math>. Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions. 

<math display="block">\begin{align} 
\Psi_{V>E} (x) \approx A \frac{ e^{\frac 2 3 u^\frac{3}{2}}}{\sqrt[4]{u}} + B \frac{ e^{-\frac 2 3 u^\frac{3}{2}} }{\sqrt[4]{u}} \\   
\Psi_{E>V}(x) \approx C \frac{\cos{(\frac 2 3 u^\frac{3}{2} - \alpha )  } }{\sqrt[4]{u} } + D \frac{ \sin{(\frac 2 3 u^\frac{3}{2} - \alpha)}}{\sqrt[4]{u} }\\   

\end{align} </math>

==== 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>

==== 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>

==== Common oscillating wavefunction ====
Matching the two solutions for region <math>x_1<x<x_2  </math>, it is required that the difference between the angles in these functions is <math>\pi(n+1/2)</math> where the <math>\frac \pi 2</math> phase difference accounts for changing cosine to sine for the wavefunction and <math>n \pi</math> difference since negation of the function can occur by letting <math>N= (-1)^n N'  </math>. Thus:
<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = (n+1/2)\pi \hbar ,</math>
Where ''n'' is a non-negative integer. This condition can also be rewritten as saying that:
::The area enclosed by the classical energy curve is <math>2\pi\hbar(n+1/2)</math>.
Either way, the condition on the energy is a version of the [[Bohr–Sommerfeld quantization]] condition, with a "[[Lagrangian Grassmannian#Maslov index|Maslov correction]]" equal to 1/2.<ref>{{harvnb|Hall|2013}} Section 15.2</ref>

It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual eigenfunction. In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator.<ref>{{harvnb|Hall|2013}} Theorem 15.8</ref> Specifically, the error in the energies is small compared to the typical spacing of the quantum energy levels. Thus, although the "old quantum theory" of Bohr and Sommerfeld was ultimately replaced by the Schrödinger equation, some vestige of that theory remains, as an approximation to the eigenvalues of the appropriate Schrödinger operator.

==== General connection conditions ====
Thus, from the two cases the connection formula is obtained at a classical turning point, <math>x=a
</math>:<ref name=":2" />

<math>  \frac{N}{\sqrt{|p(x)|}} \sin{\left(\frac 1 \hbar \int_{x}^{a} |p(x)| dx - \frac \pi 4\right)} \Longrightarrow - \frac{N}{\sqrt{|p(x)|}}\exp{\left(\frac 1 \hbar \int_{a}^{x} |p(x)| dx \right)}   </math>

and:

<math>  \frac{N'}{\sqrt{|p(x)|}} \cos{\left(\frac 1 \hbar \int_{x}^{a} |p(x)| dx - \frac \pi 4\right)} \Longleftarrow  \frac{N'}{2\sqrt{|p(x)|}}\exp{\left(-\frac 1 \hbar \int_{a}^{x} |p(x)| dx \right)}  </math>

The WKB wavefunction at the classical turning point away from it is approximated by oscillatory sine or cosine function in the classically allowed region, represented in the left and growing or decaying exponentials in the forbidden region, represented in the right. The implication follows due to the dominance of growing exponential compared to decaying exponential. Thus, the solutions of oscillating or exponential part of wavefunctions can imply the form of wavefunction on the other region of potential as well as at the associated turning point.