WKB quantization condition

For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of $\pi$ with the turning-point phase correction.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.

A Dirichlet boundary condition at a hard endpoint and a single smooth classical turning point give the leading WKB quantization condition ${\int }_0^{x_{\mathrm{t}}}k(x)dx=(n+3/4)\pi$, with $n=0,1,\dots$. The phase differs from the two-soft-turning-point condition because the hard wall imposes an exact wavefunction zero rather than an Airy function connection phase. The WKB validity condition is still checked in the interior.

For the half-line square-root potential well, the dimensionless action integral isHence $E_n/E_{\ast }=[15\pi (n+3/4)/8{]}^{2/5}$. The ground-state estimate is $1.811694E_{\ast }$. This is a semiclassical estimate, not a variational upper bound.

For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.

For the even potentialthe turning points are $\pm 4E$ for $0\le E\le 1$ and $\pm (E+1{)}^2$ for $E\ge 1$. Its action is