Question

Oscillating electron in an atom.

Light of wavelength 590 nm is emitted by an electron in an atom behaving as a lightly damped simple harmonic oscillator with a Q value of 8.0 x 10^7. From the resonance bandwidth the width of the spectral line from such an atom in units of metres is:

Answer 1

\[
\newcommand\metre{\text m}
\newcommand\U[1]{[#1]}
\newcommand\E[1]{\times10^{#1}}
\boxed{\ast}\\[3ex]
Q
%= \frac{\omega}{\Delta\omega}\\
%= \frac{\omega}{\Gamma}\\
%= \frac{\nu}{\Delta\nu}\\
= \frac{\lambda}{\Delta\lambda}\\[3ex]
\Delta\lambda\ \ \
= \frac\lambda Q\\
\qquad= \frac{590\E{-9}\U{\metre}}{8.0\E7}\\
% 8.428571429e-18
\qquad= 8.4\E{-16}\U\metre
\]

Answer 2

is this right?

Answer 3

from the theory of resonant circuit, we have the subsequent expression for the Quality factor Q:
\[\Large Q = \frac{{{\omega _0}}}{{2\Delta \omega }}\]
where \omega_zero is the resonant frequency
Now we have this:
\[\Large Q = \frac{{{\omega _0}}}{{2\Delta \omega }} = \frac{{2\pi {\nu _0}}}{{2 \cdot 2\pi \Delta \nu }} = \frac{{{\nu _0}}}{{\Delta \nu }}\]
on the other hand we can write this:
\[\large \Delta \lambda = \frac{c}{{{\nu _1}}} - \frac{c}{{{\nu _2}}} = c\frac{{{\nu _2} - {\nu _1}}}{{{\nu _1} \cdot {\nu _2}}} \cong c\frac{{\Delta \nu }}{{v_0^2}} \Rightarrow \Delta \nu = \frac{{v_0^2 \cdot \Delta \lambda }}{c}\]

Answer 4

oops..
I have made a typo:
\[\Large Q = \frac{{{\omega _0}}}{{2\Delta \omega }} = \frac{{2\pi {\nu _0}}}{{2 \cdot 2\pi \Delta \nu }} = \frac{{{\nu _0}}}{{2 \cdot \Delta \nu }}\]
so substituting into the expression for Q, we get:
\[\Large Q = \frac{{{\nu _0}}}{{2\Delta \nu }} = \frac{{{\nu _0}}}{{2 \cdot \frac{{v_0^2 \cdot \Delta \lambda }}{c}}} = \frac{{{\lambda _0}}}{{2 \cdot \Delta \lambda }}\]

Answer 5

and finally:
\[\Large \Delta \lambda = \frac{{{\lambda _0}}}{{2Q}}\]

Answer 6

oops.. I have made a typo:
please instead \[\Large {v_0}\]
read \[\Large {\nu _0}\]

Answer 7

you smart huh @michele_Laino

Answer 8

Where do you get the factor of two from, (in the denominator)?

Answer 9

@Michele_Laino

Answer 10

I have used the definition which comes from the theory of resonant circuits