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 quantum system of two spin $1/2$ is in a generix mixture of singlet and triplet states.

  • Write the full density matrix

  • write the reduced density matrix of one spin

  • evaluate for that spin the average of $\sigma_x$ and $\sigma_z$

c) 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