Editing WKB approximation (section)

==== 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.