Doppler Effect and Relativity

[aznkid310]

Homework Statement

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)

Homework 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

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)])

 

[Gokul43201]

You are told that the molecular speeds are non-relativistic, so you can either use the non-relativistic form of the Doppler equations, or simply take the limit v<<c in the above expression.

Then the thing you want to do is find the spread in wave-lengths, which is the difference between the wave-lengths observed for the 2 cases: (i) moving towards, and (ii) moving away from the observer.

 

[aznkid310]

Do you mean to take v -> 0 in the Y expression and then do it for moving toward the observer? Then take the difference?

If i to that, then Y = (sqrt[2kT/m])/f_s for both cases.

 

[Gokul43201]

Do you mean to take v -> 0 in the Y expression and then do it for moving toward the observer? Then take the difference?

Not v-> 0, but v/c << 1. You need to Taylor expand the square root quantity and discard terms of second or higher order in v/c. What you get should be the same expression as the non-relativistic equation.