2. (20 pts) We can consider the lineshape for a damped harmonic oscillator, and some other results, using the correlation function not of displacements, but rather of velocities. Consider then a harmonic oscillator without electrical anharmonicity, so that H(q) = H(qo) + (q - 90) ag to with go the equilibrium displacement. One can evaluate the lineshape from the formula I (w) = 2Re dt (u(t)u(0)> e-last If the oscillator is simply damped by interaction with its environment, the dipole will follow the Langevin equation mq = -ymq - mwa(q - 90) + R Now define x = q - qo as the displacement away from equilibrium. Then defining du/0q lo = 20, a standard charge; we can then write the velocity x in terms of / and Q0. a) Show that I (w) = - (w2 - wg)2+ 1202 b) Evaluate the integral I'(w) = 2Re dt (ji(t)(0)> e-it Write it in terms of ( (t)(0)). c) For harmonic oscillators in a canonical ensemble, the equipartition theorem can be written (= mw x2) = (mo?) = =m(x2) = 2 Use this, with your results for I(w) and for /'(w), to relate /'(w) to /(w). The result that you have found for this relationship for the special case of harmonic oscillators turns out to be true in general. d) Compute the frequency Wmax at which the lineshape maximizes. Is this the value that you expected for a damped oscillator? Comment on the relationship of lineshape position maximum to linewidth