Exercise #4

a) a quantum system of spin $1/2$ can be described as

  • a single state (quantum superposition) of $|\uparrow>$ and $\downarrow >$ having the same probability

  • an ensemble (quantum mixture) with equal probability of finding the spin up or down

In the two previous cases evaluate the average of $\sigma_x$ and $\sigma_z$

b) a classical ensemble of two one dimensional non-interacting oscillators is described by the distribution function $\rho(q_1,q_2,p_1,p_2,t)$.

  • write explicitly the Liouville’s equation of motion for $\rho(q_1,q_2,p_1,p_2,t)$

  • write explicitly the equation of motion for the reduced density $\rho(q_1,p_1,t)$

  • show that changing the variables into action/angle the distribution function with uniform angle between $[0,2\pi]$ is the stationary distribution for each oscillator

c) consider the system of a classical particle interacting with N harmonic oscillators. Assume that the potential which acts on the particle alone is also harmonic.

  • is the system integrable?

  • is the system ergodic?

  • provided the answer to the previous questions, discuss the effective equation of motion as derived in the lecture.