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

• xago
For example, try n = 2 for the first one, then n = 3, etc. Keep a table of ratios for each guess. Once you have the ratios, you can use the Rydberg equation to find a ratio of n's that gives the observed ratios. Then you can use the Rydberg equation to find the actual n's themselves. For example, if n1 = 3 and n2 = 4, then you get a ratio of 4. Use the observed ratio to get an estimate for the Rydberg constant. This is just one example. You might need to use more than one transition to get a good estimate.In summary, the spectral
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: fobs = $$\frac{\sqrt{1 + \frac{v}{c}}}{\sqrt{1 - \frac{v}{c}}}$$ * fsource

## 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

xago said:
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).

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.

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

## 1. What are initial and final quantum numbers?

Initial and final quantum numbers refer to the energy levels of an atom or molecule. The initial quantum number represents the starting energy level of an electron, while the final quantum number represents the ending energy level after a transition has occurred.

## 2. How does the Doppler shift affect spectral lines?

The Doppler shift is a change in frequency or wavelength of a wave due to the relative motion between the source of the wave and the observer. When this shift occurs in the light emitted by an object, it can cause a change in the position of its spectral lines. This shift can provide information about the velocity and direction of the object.

## 3. What is the significance of spectral lines in spectroscopy?

Spectral lines are unique patterns of light emitted or absorbed by atoms or molecules. These lines can provide information about the composition, temperature, and motion of a celestial object. In spectroscopy, scientists use these lines to identify and study the chemical and physical properties of different substances.

## 4. How are spectral lines created?

Spectral lines are created when electrons in an atom or molecule undergo a transition from a higher energy level to a lower energy level. This transition results in the emission or absorption of light at specific wavelengths, which appear as distinct lines on a spectrum.

## 5. How can spectral lines be used to determine the composition of a star?

The unique arrangement of spectral lines in the light emitted by a star can reveal its chemical composition. Each element produces a distinct pattern of spectral lines, allowing scientists to identify the elements present in a star's atmosphere. By analyzing these lines, scientists can also determine the star's temperature and other physical characteristics.

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