We have the Schrodinger eqn\begin{equation}(\Delta+V(r))R(r)=E R(r)\end{equation}where we can take $V(r)=-k/r$ for the beginning and we impose on the reduced radial function $u(r)=r R(r)$ the boundary conditions $u(0)=u(\infty)=0$.The problem is to:Solve numerically inwards and outwards the radial equation, and determine the eigenenergy by imposing the vanishing of the wronskian at some intermediate point.How to proceed?1) What is meant by solving inwards and outwards and what does imposing the vanishing of the wronskian at some intermediate point mean for us? 2) What numerical method shall I use? The Numerov method is probably not applicable, right?
Solve numerically inwards and outwards the radial equation?
homework;quantum mechanics
This is what brought me meanwhile to the solution:1) Solving inwards and outwards simply means numerically integrating the mentioned differential equation:from some small $r_1$ outwards to some $r_2>r_1$ (suitable BC in our case: $u_{out}(r_1)=r_1, u_{out}'(r_1)=1$ for example) and from some larger $r_3>r_2$ inwards back to $r_2$ (suitable BC for example: $u_{in}(r_3)= e^{-\sqrt{E} r_3}, u_{in}'(r_3) = -\sqrt{E} u(r_3)$, as we expect exponential decay)2) If the wronskian $$W(E):=u_{out}'(r_2) u_{in}(r_2) - u_{out}(r_2) u_{in}'(r_2)$$vanishes at point $r_2$, it means that the two solutions are linerly dependent and we get the energy eigenvalue(s) $E$ we were looking for (we have to find the roots of the function W(E)=0).we expect exponential decay, 3) In Mathematica, for instance, the ImplicitRungeKutta method for solving the equation for $u_{out}$ and $u_{in}$, i.e. $$-u''(r) - u(r)/r= -E u(r)$$ seems to be quite suitable.