Editing WKB approximation (section)

===Approximation away from the turning points===
The wavefunction can be rewritten as the exponential of another function {{math|S}} (closely related to the [[Action (physics)|action]]), which could be complex,
<math display="block">\Psi(\mathbf x) = e^{i S(\mathbf{x}) \over \hbar},    </math>
so that its substitution in Schrödinger's equation gives:

<math display="block">i\hbar \nabla^2 S(\mathbf x) - (\nabla S(\mathbf x))^2 = 2m \left( V(\mathbf x) - E \right),</math>

Next, the semiclassical approximation is used. This means that each function is expanded as a [[power series]] in {{mvar|ħ}}.
<math display="block">S = S_0 + \hbar S_1 + \hbar^2 S_2 + \cdots </math>
Substituting in the equation, and only retaining terms up to first order in {{math|ℏ}}, we get:
<math display="block">(\nabla S_0+\hbar \nabla S_1)^2-i\hbar(\nabla^2 S_0) = 2m(E-V(\mathbf x)) </math>
which gives the following two relations:
<math display="block">\begin{align}
(\nabla S_0)^2= 2m (E-V(\mathbf x)) = (p(\mathbf x))^2\\ 
2\nabla S_0 \cdot \nabla S_1 - i \nabla^2 S_0 = 0 
\end{align}</math>
which can be solved for 1D systems, first equation resulting in:<math display="block">S_0(x) = \pm \int \sqrt{ 2m \left( E - V(x)\right) } \,dx=\pm\int p(x) \,dx     </math>and the second equation computed for the possible values of the above, is generally expressed as:<math display="block">\Psi(x) \approx C_+ \frac{ e^{+ \frac i \hbar \int p(x)\,dx} }{\sqrt{|p(x)| }} + C_- \frac{ e^{- \frac i \hbar \int p(x)\,dx} }{\sqrt{|p(x)| }} </math>


Thus, the resulting wavefunction in first order WKB approximation is presented as,<ref>{{harvnb|Hall|2013}} Section 15.4</ref><ref name=":1">{{Cite book |last=Zettili |first=Nouredine |title=Quantum mechanics: concepts and applications |date=2009 |publisher=Wiley |isbn=978-0-470-02679-3 |edition=2nd |location=Chichester}}</ref>
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|equation =  <math>\Psi(x)    \approx \frac{ C_{+} e^{+    \frac{i}{\hbar}  \int \sqrt{2m \left( E -  V(x)  \right)}\,dx} +  C_{-} e^{- \frac{i}{\hbar} \int \sqrt{2 m \left( E - V(x)  \right)}\,dx} }{  \sqrt[4]{2m \mid E - V(x) \mid} }  </math>
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In the classically allowed region, namely the region where <math>V(x) < E</math> the integrand in the exponent is imaginary and the approximate wave function is oscillatory. In the classically forbidden region <math>V(x) > E</math>, the solutions are growing or decaying. It is evident in the denominator that both of these approximate solutions become singular near the classical '''turning points''', where {{math|1=''E'' = ''V''(''x'')}}, and cannot be valid. (The turning points are the points where the classical particle changes direction.)


Hence, when <math>E > V(x)</math>, the wavefunction can be chosen to be expressed as:<math display="block">\Psi(x') \approx C \frac{\cos{(\frac 1 \hbar \int |p(x)|\,dx} + \alpha)  }{\sqrt{|p(x)| }} + D \frac{ \sin{(- \frac 1 \hbar \int |p(x)|\,dx} +\alpha)}{\sqrt{|p(x)| }}  </math>and for <math>V(x) > E</math>,<math display="block">\Psi(x')    \approx \frac{ C_{+} e^{+    \frac{i}{\hbar}  \int |p(x)|\,dx}}{\sqrt{|p(x)|}} + \frac{ C_{-} e^{- \frac{i}{\hbar} \int |p(x)|\,dx} }{ \sqrt{|p(x)|} } . </math>The integration in this solution is computed between the classical turning point and the arbitrary position x'.