Exercise #3

a) Consider harmonic oscillators of frequency $\omega}$ in two distinct cases

  1. An ensemble of thermalized oscillators described by the density matrix $\hat{\rho}=\sum_n P_n |n><n|$ where $|n>$ are eigenstates of harmonic oscillators and $P_n\propto \exp(-\beta E_n)$

  2. A pure state $|\psi>=\sum_n \sqrt{P_n}|n>$ with the same $P_n$ as in i)

In both case calculate $<x^2>$ as a function of temperature - assuming a Boltzmann distribution for $P_n$ - and comment the results.

b) Consider N classical independent one-dimensional harmonic oscillators (mass $m_i$, frequency $\omega_i$ $i=1,N$) whose initial data are taken according Boltzmann distribution for position and momenta at temperature $T$. Let them evolve and calculate the following correlation functions:

$C_{i,j}(t,t')=< p_i(t)p_j(t') >$, $G_{i,j}(t,t')=< x_i(t)x_j(t') >$ where averages and done on the Boltzmann distribution of initial data.

By assuming a distribution of the oscillator’s frequencies $P(\omega)\propto \omega^2 e^{-\omega/\Lambda}$ ($\omega_i>0$) as well as equal masses for all oscillators calculate:

$C(t,t')=(1/N)\sum_{i,j} C_{i,j}(t,t')$, $G(t,t')=(1/N)\sum_{i,j} G_{i,j}(t,t')$

find the time evolution of these functions, temperature dependence and dependence on the exponential distribution parameter.

hint : express the solution for x and p as a function of initial data x(0) and p(0).