Initial and final quantum numbers/Doppler shift/Spectral lines Question

[xago]

Homework Statement

The spectral lines of the light from distant galaxies exhibit a Doppler redshift due to
the motion of the galaxy away from our own. For a particular galaxy, three spectral
absorption lines are observed from the Earth to have wavelengths of 164.4 nm 168.4 nm
and 177.5 nm. Assuming that these lines are from hydrogen gas in the outer regions
of the stars in the galaxy, answer the following:

(a) Identify the initial and final quantum numbers for the states involved in these
three transitions.

(b) Determine the speed with which the galaxy is receding from our own.

Homework Equations

a)Rydberg Equation: \frac{1}{\lambda} = R*(\frac{1}{nf^2} - \frac{1}{ni^2})

b)Relativistic Doppler Shift: f_obs = \frac{\sqrt{1 + \frac{v}{c}}}{\sqrt{1 - \frac{v}{c}}} * f_source

The Attempt at a Solution

For a) since were given 3 wavelengths I've simplified it down to 1/\lambda/R = (\frac{1}{nf^2} - \frac{1}{ni^2}). The probelm is, when I plug in the wavelength of 164.3nm, it simplifies to 0.5543 = (\frac{1}{nf^2} - \frac{1}{ni^2}). Now I've got 2 unknowns and 1 equation. If i just sub in whole integers for ni and nf to try and find the best combination, I still can't come close to 0.5543.

For b) I was thinking just to use the realativistic Doppler Shift equation, but I'm only given the observed wavelength and not the wavelength of the source or the velocity between the two. Once again, 2 unknowns, 1 equation.

If anyone could tell me what I'm missing for the 2 questions I'd be forever grateful

 

[kuruman]

For a) since were given 3 wavelengths I've simplified it down to 1/\lambda/R = (\frac{1}{nf^2} - \frac{1}{ni^2}). The probelm is, when I plug in the wavelength of 164.3nm, it simplifies to 0.5543 = (\frac{1}{nf^2} - \frac{1}{ni^2}).

You have to understand that you just can't plug in and expect a wavelength match because the wavelength you are comparing against is Doppler-shifted by being multiplied by some factor related to the velocity. You need to drop the Doppler factor out of the picture, then do your educated guesswork. What would happen if you calculated ratios of wavelengths? They are independent of the Doppler shift, are they not? What does the ratio of any two wavelengths look like according to the Rydberg formula? Also note that the observed wavelengths are in the UV region, so I would try transitions to the n = 1 state first (Lyman series).

 

[xago]

Thanks for your response, I see what you are saying about taking ratios to elimiate the redshift factor of the observed wavelengths. By taking the wavelength ratios and simplifying a bit I get:
\frac{\lambda1}{\lambda2} = \frac{1/nf2^2-1/ni2^2}{1/nf1^2-1/ni1^2}

Now my question is should I assume that they both drop to the n=1 level in which nf2² and nf1² become 1 and therefore:
\frac{\lambda1}{\lambda2} = \frac{1-1/ni2^2}{1-1/ni1^2}

If so, then I still get an equation with 2 unknowns and the guesswork here seems to punishing to do.

 

[kuruman]

It's not that bad. Take the ratios of the given wavelengths first, then try the first few values for the n's.