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

Hydrogen's only visible spectral lines are 656, 486, 434, and 410 nm. If spectral lines were of absolutely precise wavelength, they would be very difficult to discern. Fortunately, two factors broaden them: the uncertainty principle and the doppler broadening.

Atoms in a gas are in motion, so some light will arrice that was emitted by the atoms moving toward the observer and some atoms moving away. Thus, the light reaching the observer will cover the range of wavelengths.

Making the assumption that atoms move no faster than their rms speed

v_r = sqrt[2kT/m], where k = boltzman constant

Obtain a formula for the range of wavelengths in terms of the wavelength Y of the spectral line, the atomic mass m, and the temperature T. (note v_r << c)

2. Relevant equations

I'm not sure how to solve for a whole range of wavelengths. I used the formula Y = v/f and plugged in the doppler formula, but I am stuck there.

The resulting answer should be { 2Y*sqrt[(3kT/m)] }/c

3. The attempt at a solution

Moving toward observer:

f_observed = (f_source)*sqrt[(1+v/c)/(1-v/c)]

Then Y = v/f_observed = (sqrt[2kT/m]) / ((f_source)*sqrt[(1+v/c)/(1-v/c)])

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# Doppler Effect and Relativity

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