The relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass $m$ under the influence of the central potential

$$V(r)=\left\{\begin{array}{rr} -U & r<a \\ 0 & r>a \end{array}\right.$$

where $U$ and $a$ are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and a neutron, giving the condition on $U$ for this state to exist.

[If $\psi$ is spherically symmetric then $\nabla^{2} \psi=\frac{1}{r} \frac{d^{2}}{d r^{2}}(r \psi)$.]