WKB approximation: Difference between revisions

Revision as of 08:21, 15 April 2025

Revision as of 00:36, 20 April 2025

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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{ \frac{2m}{\hbar^2} \left( E - V(x)\right) } \,dx=\pm\frac{1}{\hbar}\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>