Past exam of the further quantum theory course of the University of Oxford / 2023 / 3 / a

In the central classically forbidden region, define the positive decay momentum $\chi (x)=\sqrt{2m[V(x)-E]}$ and the dimensionless action integral $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi (y)dy$. The general leading WKB approximation isBoth coefficients are allowed: this interval has finite length, so the exponential that grows towards the barrier centre cannot be discarded by a condition at infinity. The WKB validity condition holds away from the classical turning point $x_1$; a local Airy function solution supplies its continuation there.

In the right-hand classically allowed region, put $p(x)=\sqrt{2m[E-V(x)]}$ and $s(x)={\hslash }^{-1}{\int }_{x_1}^{x}p(y)dy$. A convenient form of the general WKB approximation isThe phase origin $\pi /4$ is a choice of basis for the two oscillatory solutions, anticipating the Airy turning-point connection formula. Equivalently, one may use independent $e^{is}$ and $e^{-is}$ branches. The coefficients have not yet been related to those in either forbidden region.

Beyond the outer classical turning point, let $\chi (x)=\sqrt{2m[V(x)-E]}$ and $U(x)={\hslash }^{-1}{\int }_{x_2}^x\chi (y)dy$. The bound state condition removes the growing branch at positive infinity, leavingHere $F$ is an arbitrary common amplitude pending the matching and wavefunction normalization. For the confining potential energy in the native diagram, the decaying branch gives the required exterior square-integrable function.