Formulas re quantised energy of photons emitted by electrons

Question

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!

$\begingroup$ What is Z in your second formula? $\endgroup$
– A Nejati
Commented Sep 28, 2018 at 19:50 — A Nejati, Commented Sep 28, 2018 at 19:50
$\begingroup$ @AlNejati It's the Atomic Number $\endgroup$
– Pregunto
Commented Sep 28, 2018 at 20:14 — Pregunto, Commented Sep 28, 2018 at 20:14

HiddenBabel · Accepted Answer · 2018-09-28 20:13:14Z

1

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

answered Sep 28, 2018 at 20:13

HiddenBabel

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$\begingroup$ Thanks! I read the link, but I am still not sure about the correct interpretation. Rh/hc is referred to as R(infinity), but they seem to be used in different contexts. Also, I could not find hc in the definition of R*, so what do you mean when you say that it takes it in front? $\endgroup$
– Pregunto
Commented Sep 28, 2018 at 20:25
$\begingroup$ 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 $hc/h^3c$ $\endgroup$
– HiddenBabel
Commented Sep 28, 2018 at 20:30

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