Editing WKB approximation (section)

=== Validity of WKB solutions ===
From the condition:<math display="block">(S_0'(x))^2-(p(x))^2 + \hbar (2 S_0'(x)S_1'(x)-iS_0''(x)) = 0 </math>

It follows that: <math display="inline">\hbar\mid 2 S_0'(x)S_1'(x)\mid+\hbar \mid i S_0''(x)\mid \ll \mid(S_0'(x))^2\mid +\mid (p(x))^2\mid </math>


For which the following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation:

<math display="block">\begin{align}
\hbar \mid S_0''(x)\mid \ll \mid(S_0'(x))^2\mid\\
2\hbar \mid S_0'S_1' \mid \ll \mid(p'(x))^2\mid
\end{align}    </math>

The first inequality can be used to show the following:

<math display="block">\begin{align}
\hbar \mid S_0''(x)\mid \ll \mid(p(x))\mid^2\\
\frac{1}{2}\frac{\hbar}{|p(x)|}\left|\frac{dp^2}{dx}\right| \ll |p(x)|^2\\
\lambda \left|\frac{dV}{dx}\right| \ll \frac{|p|^2}{m}\\
\end{align}    </math>

where <math display="inline">|S_0'(x)|= |p(x)|    </math> is used and <math display="inline">\lambda(x) </math> is the local [[De Broglie waves|de Broglie wavelength]] of the wavefunction. The inequality implies that the variation of potential is assumed to be slowly varying.<ref name=":1" /><ref name=":2">{{Cite web |last=Zwiebach |first=Barton |title=Semiclassical approximation |url=https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/bf207c35150e1f5d93ef05d4664f406d_MIT8_06S18ch3.pdf}}</ref> This condition can also be restated as the fractional change of <math display="inline">E-V(x) </math> or that of the momentum <math display="inline">p(x) </math>, over the wavelength <math display="inline">\lambda </math>, being much smaller than <math display="inline">1 </math>.<ref>{{Cite book |last1=Bransden |first1=B. H. |url=https://books.google.com/books?id=ST_DwIGZeTQC |title=Physics of Atoms and Molecules |last2=Joachain |first2=Charles Jean |date=2003 |publisher=Prentice Hall |isbn=978-0-582-35692-4 |pages=140–141 |language=en}}</ref>



Similarly it can be shown that <math display="inline">\lambda(x) </math> also has restrictions based on underlying assumptions for the WKB approximation that:<math display="block">\left|\frac{d\lambda}{dx}\right| \ll 1 </math>which implies that the [[De Broglie waves|de Broglie wavelength]] of the particle is slowly varying.<ref name=":2" />