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## Homework Statement

The emission wavelengths of hydrogen-like atoms are related to nuclear charge. How do they scale as a function of Z? What are the longest and shortest wavelengths in the Balmer series for ##Li^{2+}##?

## Homework Equations

##E_n = -\frac{R}{n^2}## (1)

##a_0 = \frac{\hbar^2}{Zme^2}## (2)

From (2), ##\hbar^2 = a_0Zme^2## (3)

##\alpha = \frac{Ze^2}{\hbar c}## (4)

##R = \frac{1}{2} mc^2 \alpha^2## (5)

## The Attempt at a Solution

I started by trying to find an expression for the energy levels of hydrogen-like atoms. Substituting (5) into (1) gives

##E_n = -\frac{mc^2\alpha^2}{2n^2}##

Substitute in (4):

##E_n = -\frac{mZ^2e^4}{2\hbar^2 n^2}##

Sub in (3):

##E_n = -\frac{mZ^2e^4}{2Za_0me^2 n^2}##

Simplifying, ##E_n = -\frac{Ze^2}{2a_0n^2}##. This can be related to wavlength via ##E_n = \frac{hc}{\lambda}##, so emission and absorption wavelengths of hydrogenic atoms are related to nuclear charge Z by the function

##\lambda = -\frac{2hca_0n^2}{Ze^2}##

But I have no idea how to answer the bit about the Balmer series because n=2 and Z=3, so based on my equation isn't there only going to be one answer? How do I get minimum/maximum wavelengths?

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