The Schrodinger equation for a local, central potential

The kinetic-energy operator separates in spherical polar coordinates into radial and angular observables given by (3.63). The Schrodinger equation for a local, central potential is therefore 

We can rewrite the radial equation in terms of a simpler differential operator than (3.64) by solving for the function 

We choose a particular eigenstate (3.65) lVm(r) of and replace L by its eigenvalue, obtaining the radial equation 

If E is negative, (4.10) is an eigenvalue problem with solutions u r) and eigenvalues e . If E is positive the solution with the correct boundary conditions is a linear combination of angular-momentum eigenstates. 

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