The one-electron central-field system

Let us consider in general terms the quantum-mechanical treatment of one-electron systems in a central field. Such systems are treated in detail in most textbooks on quantum mechanics see, for example, [9]. Our emphasis here is on the separation of the solution into radial and angular 

For an electron in a central field, the Schrfidinger equation may be written as 

In polar coordinates, the Laplacian in the Schrodinger equation (6.3.1) takes the form 

All angular dependencies are thus isolated in the angular-momentum operator and there is no term in the Hamiltonian that involves differentiation with respect to both the radial and the angular coordinates. We therefore attempt a solution to the Schrd nger equation in the product form 

With the angular part in the form of a spherical harmonic, we find that the radial part may be obtained from the differential equation 

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