(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

By expanding a MacLaurin Series show that

[tex]E_{n}=\epsilon_{n} - \mu c^{2} = - \frac{w_{0}Z^{2}}{n^{2}}[1+\frac{\alpha^{2} Z^{2}}{n}(\frac{1}{k}-\frac{3}{4n})][/tex]

2. Relevant equations

Through a lengthy derivation I arrived at

[tex]\epsilon_{n}=\frac{\mu c^2}{\sqrt{1+\frac{Z^{2}\alpha^{2}}{n_{r}+\sqrt{l^{2}-Z^{2}\alpha^{2}}}}}[/tex]

I should add that k is what the text is using for the azimuthal quantum number, I used l in my derivation out of habit.

3. The attempt at a solution

I've got no ideas where to go with this thing. I should take advantage of identites

[tex]\sqrt{1-x}=1-\frac{x}{2}-\frac{x^{2}}{8}+...[/tex]

[tex]\frac{1}{1+x}=1+...[/tex]

Do I need to make some aggressive substitutions?

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# Relativistic Bohr Atom and MacLaurin Series

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