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For the equation
<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>
with {{math|''Q''(''x'') <0}} an [[analytic function]], the value <math>n_\max</math> and the magnitude of the last term can be estimated as follows:<ref>{{cite journal| last=Winitzki |first=S. |year=2005 |arxiv=gr-qc/0510001 |title=Cosmological particle production and the precision of the WKB approximation |journal=Phys. Rev. D |volume=72 |issue=10 |pages=104011, 14 pp |doi=10.1103/PhysRevD.72.104011 |bibcode = 2005PhRvD..72j4011W |s2cid=119152049 }}</ref>
<math display="block">n_\max \approx 2\epsilon^{-1} \left| \int_{x_0}^{x_{\ast}} \sqrt{-Q(z)}\,dz \right| , </math>
<math display="block">\delta^{n_\max}S_{n_\max}(x_0) \approx \sqrt{\frac{2\pi}{n_\max}} \exp[-n_\max], </math>
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the number {{math|''n''<sub>max</sub>}} will be large, and the minimum error of the asymptotic series will be exponentially small.
== Application in non
[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the turning points]]
[[File:WKB approximation to probability density.svg|thumb|Probability density for the approximate wave function. Vertical lines show the turning points]]
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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^{
=== Validity of WKB solutions ===
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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 ====
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<math display="block">E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. </math>
This result is also consistent with the use of equation from [[bound state]] of one rigid wall without needing to consider an alternative potential.
=== Quantum Tunneling ===
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