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## Homework Statement

For an atom X, the high-n levels have energies ##E_n = -\mu \frac{(\alpha c)^2}{n^2}## with ##\alpha = \frac{e^2}{\hbar c}##

Find the frequency shift ##\nu_{Hen\alpha} - \nu_{Hn\alpha}## for ##n\alpha## giving a transition frequency near 142MHz

(The notation here means that the electron moves by one energy level)

## Homework Equations

## The Attempt at a Solution

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##E_f - E_i = -\mu \frac{(\alpha c)^2}{(n-1)^2} + \mu \frac{(\alpha c)^2}{n^2}##

## = \mu (\alpha c)^2 (\frac{1}{n^2} - \frac{1}{(n-1)^2}) = h\nu##

rearranging and simplifying leads to

##\frac{1-2n}{(n-1)^2 n^2} = \frac{h \nu}{\mu (\alpha c)^2}##

Using a simplification I found justified here: http://www.cv.nrao.edu/course/astr534/Recombination.html

##\frac{1-2n}{(n-1)^2 n^2} \approx \frac{2}{n^3}##

which leads to

##n \approx (\frac{2 \mu (\alpha c)^2}{h \nu})^{\frac{1}{3}}##

solving for n using the value of the fine structure constant given in the question (which is in cgs units I believe), I get nonsense answers of ##n < 1##.

Using the value of ##\alpha = \frac{ke^2}{\hbar c}## in SI units, I get values of n that produce the correct frequency when plugged back into my formula, but do not agree with radio combination lines that I looked up here: http://adsabs.harvard.edu/full/1968ApJS...16..143L

For example, using the fine structure constant in SI units, for Hydrogen I find ##n = 452##, which when plugged into

##\nu = \frac{\mu (\alpha c)^2}{h} (\frac{1}{n^2} - \frac{1}{(n-1)^2})##

I get ##\nu = 142.5MHz##, but from the above link I find that the transition should correspond to n=359.

I'm really struggling to get my head around where the discrepancy lies.

Thank you in advance for any help you can give!

EDIT - Ok, so I was using the wrong units for ##e## which solves my concerns about the fine structure's value, which is dimensionless. However, I am still stuck regarding the discrepancy between my calculated n values, and those I have found in the literature.

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