The central force problem involves solving the Schrodinger equation for a particle moving under the influence of a central potential, which is a spherically symmetric function.
The Hamiltonian of a particle subject to a central force is
where (see this article for derivation) and
is a function of
only since it is a spherically symmetric function.
Substituting eq49 and eq50 in eq300 gives
The eigenfunctions of , which is the operator for the square of the magnitude of the orbital angular momentum of the particle, are the spherical harmonics
, which are independent of
(for molecules, we use
and
instead of the quantum numbers
and
). Therefore, the Hamiltonian is separable and the solution to the Schrodinger equation is of the form
In conclusion, the Schrodinger equation of a particle subject to a central force separates into radial and angular parts. This is possible because of the spherically symmetric potential. The eigenfunction is the product of two functions, each independent of the other’s coordinates. The central force problem arises when solving the Schrodinger equations for the hydrogen atom and the nuclear motion of a diatomic molecule.