Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 2 / a / Solution

Let $b=a/2$, $q=\sqrt{2m(U_0-|E|)}/\hslash$ and $\kappa =\sqrt{2m|E|}/\hslash$. Bound levels have $-U_0<E<0$. Up to normalization, parity givesThe respective inverse squared normalization constants are $b+\sin (2qb)/(2q)+{\cos }^2(qb)/\kappa$ and $b-\sin (2qb)/(2q)+{\sin }^2(qb)/\kappa$. These include both exponential tails.