The Hamiltonian is

Separate the radial and angular part. The radial equation is

rearrange, using Atomic unit

Since the normalization condition of is , it is natural to define

The radial equation becomes

Substitute the eigen energy

Set

The radial equation becomes

Take out the

Now, we know the short and long-range behaviour of , assume it to be

Then the equation of is

This is the Laguerre equation

where and . Therefore, the solution of the radial equation is,

where is normalization factor.

Notice that the Laguerre polynomial is only defined for , thus, .

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