Bound states in a local, central potential

In atomic physics we encounter one-electron bound states with different types of boundary condition. For the first type the electron is completely confined to a spherical box, near the boundary of which the potential is negligible. The second type involves a potential which falls to zero 

We consider a spherical potential which is zero up to r = a, where it is infinite. The radial boundary condition results in the radial equation (4.10) being an eigenvalue problem with eigenvalues which are positive with respect to the zero of energy. We define a wave number k by 

Since (4.12) is a second-order equation has two independent forms, which are most easily understood when = 0 (s-state). In this case 

Uno f) = sin knor or cos k Qr. For the sine solution the radial wave function 

The generalisation of u o r) to arbitrary positive integers tf is given by the regular and irregular Ricatti—Bessel functions U( p) and V( p) respectively. They satisfy (4.12) with 

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