Exploring the Doppler Effect with a Laser on a Spring

[Seedling]

Homework Statement

A laser of frequency f₀ is connected to a spring which oscillates in one dimension with a period T and amplitude A. The laser excites the 1st four Paschen series lines of hydrogen. What is f₀

Homework Equations

I'm assuming that I'll need the Doppler shift equations,

f = f₀[ (1 + β) / (1 - β) ]^1/2 (for approaching)

and

f = f₀[ (1 - β) / (1 + β) ]^1/2 (for receding)

Where β = v/c

The Attempt at a Solution

I've identified the Paschen series for hydrogen to be 1870 nm, 1280 nm, 1090 nm, and 1000 nm.

I don't understand where I should go next. Initially I thought that I could plug values into the equation and solve for the original frequency but I don't see what values I could put in. For an oscillating spring, there will be infinity values of the velocity from 0 to whatever the maximum velocity is. So I don't see how I could solve it?

Part 2 of the problem also asks what the maximum velocity of this oscillation must be to give the observed lines.

Any suggestions are much appreciated.

 

[diazona]

Unless the problem is referring to some sort of weird quantum system (and I don't think it is), it probably means that there are four particular values of the velocity between -v_max and v_max at which the Doppler-shifted laser frequency (or wavelength, if you prefer) corresponds to one of the first four Paschen lines. So you basically have some information - not complete information, but still some information - about the range over which the Doppler-shifted frequency varies. The unshifted laser frequency also has to be somewhere within that range, right? Think about that.

I wonder if you can also assume that no other hydrogen emission lines are excited. That would give you an upper limit on the size of the range of shifted frequencies...

 

[Dawei]

I wonder if you can also assume that no other hydrogen emission lines are excited. That would give you an upper limit on the size of the range of shifted frequencies...

Sorry, yes, it is stated in the problem that no other lines are excited.

there are four particular values of the velocity between -vmax and vmax at which the Doppler-shifted laser frequency (or wavelength, if you prefer) corresponds to one of the first four Paschen lines. So you basically have some information - not complete information, but still some information - about the range over which the Doppler-shifted frequency varies. The unshifted laser frequency also has to be somewhere within that range, right? Think about that.

So the range of wavelengths is 870 nm, if the four lines are 1870, 1280, 1090 and 1000.

Would I just assume that the original frequency is exactly in the center of this range? That way it would shift to the highest and lowest frequencies at the positive and negative maximum velocities?

Then to calculate what this maximum velocity would need to be, I just take the 1000 nm (or the 1870 nm) and plug it into the Doppler equation?

 

[Seedling]

Would I just assume that the original frequency is exactly in the center of this range? That way it would shift to the highest and lowest frequencies at the positive and negative maximum velocities?

Then to calculate what this maximum velocity would need to be, I just take the 1000 nm (or the 1870 nm) and plug it into the Doppler equation?

OK, doing this, I get a speed of 0.29C for the velocity of the spring--i.e., much faster than what could ever be realistically achieved.

What am I missing here? There has to be something. A monochromatic laser going back and forth, how is it capable of reaching such a wide range of frequencies? There has to be something more to this that I'm not seeing