(Solved): 2) Absorption coefficient of white light: using fluctuation dissipation theorem The calculation fro ...

2) Absorption coefficient of white light: using fluctuation dissipation theorem The calculation from exercise 1 can be performed much quicker using the FDT: $$?(t)=??(t)?\frac{d}{dt}C(t),$$ where the response function splits into even and odd part as $$?(t)={?}_o(t)+{?}_e(t)$$ and the correlation function is defined as $$C(t)=?x(0)x(t)?$$. a) Show that $${?}_o(t)=?\frac{?}{2}\frac{d}{dt}C(t)$$ b) Using the result from (a) show that $${?}_{??}^{?}d??(?)=?{??\frac{d^2}{d{t}^2}C(t)|}_{t=0}.$$ c) (1P) Show that $$\frac{d^2}{d{t}^2}C(t)=?C_{vv}(t)=??\stackrel{?}{x}(0)\stackrel{?}{x}(t)?$$. d) (1P) Combine the results of (c) and (d) to obtain the final result for $$\stackrel{?}{?}$$. e) (3P) Compute the line center frequency, i.e the frequency with the maximal absorption $${?}^{?}$$, the the spectral height $$?\left({?}^{?}\right)$$ and the full width at half maximum (FWHM) $$??={?}_R?{?}_L$$ (with $$\frac{?\left({?}_L\right)}{?\left({?}^{?}\right)}=\frac{1}{2}=\frac{?\left({?}_R\right)}{?\left({?}^{?}\right)}$$ ). Interpret your result for $$\stackrel{?}{?}$$ in connection with the spectral height $$?\left({?}^{?}\right)$$ and the FWHM $$??$$. Hint: Again use the hint from $$1(d)$$ to simplify the expression $$\left(??=\sqrt{(??{)}^2}\right)$$. f) (2P) Consider a spectroscopic measurement giving a line center frequency $${?}^{?}$$, a line height $$?\left({?}^{?}\right)$$ and a FWHM $$??$$. Compute the static dielectric constant $$?={?}^{?}(0)$$ for a damped harmonic oscillator. Write the result in terms of $${?}^{?},?\left({?}^{?}\right)$$ and $$??$$. Now consider the absorption peak of water at $${?}^{?}=20\mathrm{GHz}$$ with a height of roughly $$?\left({?}^{?}\right)=8?10^{11}$$ and a width of roughly $$??=20\mathrm{GHz}$$. Estimate the static dielectric constant of water with the given constants. Why are the other spectroscopic features of water at higher frequencies rather unimportant?