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Why is the separation constant for the radial equation of the Hydrogen atom in the form of L(L+1)?

Hello everyone,

I know that solving the Schrodinger equation for the Hydrogen atom requires separation of variables. When doing this, why are the separation constants of a particular form? E.g. the separation constant for the radial equation is l(l+1), rather than just, say, l. What is the reason for this?

Thanks.
gpt
If you write the Laplacian in spherical coordinates, you find that the angular part is proportional to the **L**^(2) operator. And from the angular momentum algebra, you know that **L**^(2)|L M> = ~~h~~^(2)L(L+1)|L M>.

Since the Hamiltonian for the hydrogen atom has spherical symmetry, the energy eigenstates are also angular momentum eigenstates.

So when you separate the TDSE into the angular and radial parts, the angular part works out to that constant, and so the radial part must also equal that constant.