# Allowed energies for electrons in hydrogenic atoms

#### manenbu

I have a question which may seem stupid, but I think I missing something here.

I see 2 equations describing allowed energies for electrons in hydrogenic atoms, being:
$$E = -\frac{hcRZ^2}{n^2}$$
And
$$E = -\frac{RZ^2}{n^2}$$
I assume that both are correct, but what makes the difference? Is it the same R for both of them? or the second R includes the h and c inside?
How would I tell the difference?

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#### SpectraCat

I have a question which may seem stupid, but I think I missing something here.

I see 2 equations describing allowed energies for electrons in hydrogenic atoms, being:
$$E = -\frac{hcRZ^2}{n^2}$$
And
$$E = -\frac{RZ^2}{n^2}$$
I assume that both are correct, but what makes the difference? Is it the same R for both of them? or the second R includes the h and c inside?
How would I tell the difference?

They are trying to express the same relation, and you are correct that in the second version, the h and c have been "absorbed" into the R. In the first case the dimension of R is 1/length, and R comes from the Rydberg formula for the wavelength of a photon coupling two H-atom energy levels. In the second case, the dimension of R is energy, and the R stands for the Rydberg unit of energy, which is defined as the ground state energy of the H-atom in the approximation of infinite nuclear mass. The atomic unit of energy, the hartree, is exactly two Rydbergs.

#### manenbu

Ok, thank you.
It's just that some textbooks use either one, and none explain the difference.

Now I understand!