Theory.

1. Suppose you have a complex-valued function with the prototype

   complex F(complex);
   

   and you want to solve the complex equation F(z)=0 with the help of gsl_multiroots package. What will be your gsl_multiroot_function?

Practice.

1. Find the ground state energy and the radial wave-function of the hydrogen atom using the shooting method for solving a boundary value problem.

   Introduction:

   The s-wave radial Schrödinger equation for the Hydrogen atom reads (in units "Bohr radius" and "Hartree"),

   -(1/2)f'' -(1/r)f = εf ,

   where f(r) is the radial wave-function, ε is the energy, and primes denote the derivative over r.

   The bound s-state wave-function satisfies this equation and the two boundary conditions,

   f(r → 0) = r-r², (prove this)
   f(r → ∞) = 0 .

   These two boundary conditions can only be satisfied for certain discrete values of the energy.

   Since one cannot integrate numerically to ∞ one substitutes ∞ with a reasonably large number, rmax, such that it is much larger than the typical size of the hydrogen atom but still managable for the numerical inregrator (say, rmax = 10 Bohr radii),

   f(rmax)=0 .

   Let F_ε(r) be the solution (to be found numericall via gsl_odeiv) to our differential equation with energy ε and initial condition F_ε(r → 0)=r-r². Generally, for a random negative ε, this solution will not satisfy the boundary condition at rmax. It will only be satisfied when ε is equal one of the bound state energies of the system.

   Now define an auxiliary function

   M(ε) ≡ F_ε(rmax) .

   The shooting method is then equivalent to finding the root of the equation

   M(ε) = 0 .

   (Optional) Try also to use a more precise boundary condition for bound states (which have ε<0),

   f(r → ∞) = r e^-kr , (prove this)

   where k=√(-2ε). This should allow you to use a smaller rmax.