Formulas re quantised energy of photons emitted by electrons

I am confused about the difference between two formulas I have come across in relation to the quantised Energy of photons emitted by electrons.

$$E_{mn}=hcR_H(\frac{1}{m^2}-\frac{1}{n^2})$$

$$E_{mn}=R*(\frac{1}{m^2}-\frac{1}{n^2})$$, where $$R*=\frac{Z^2me^4}{8ε^2_0h^2}$$

$$R_H$$ refers to the R* of Hydrogen, so essential the first formula then comes out as:

$$E_{mn}=hcR*(\frac{1}{m^2}-\frac{1}{n^2})$$

Clearly this does not make sense (as the two formulas now contradict each other) and I am missing some piece of information/interpretation…

Can anybody help out?

Thanks!

• What is Z in your second formula? – Al Nejati Sep 28 '18 at 19:50
• @AlNejati It's the Atomic Number – Pregunto Sep 28 '18 at 20:14

The value of $$R_H$$ has an extra $$hc$$ in the denominator, so $$R^*$$ takes the $$hc$$ in front and simplifies the fraction. See the Wiki page... https://en.m.wikipedia.org/wiki/Rydberg_constant
• Look at the units. The way $R_H$ is defined there is in units of inverse meters. See how it has $h^3c$ in the denominator? To convert to energy, you multiply by $hc$ which has units of energy times meters. Your $R^*$ is in units of energy already, because the denominator only has $h^2$ in it, which is $hc/h^3c$ – HiddenBabel Sep 28 '18 at 20:30