4. Consider two spin-1/2 systems interacting in such a way that the Hamiltonian has the form H(12) = e Z8 Z. Suppose that |V(12) (0)) = |r4x+). Write the unitary evolution of the system and the state |V(12)(t)) at time t. Let t, = t2 = 2t1. (a) Is |V(12)(t)) entangled? (b) Is |V(12) (t2)) entangled?

2) ( Consider an electron trapped in a one-dimensional anharmonic potential. The Hamiltonian for this system is given as: H mw?£? + 2m + B&3 . Regarding the cubic term as being small, apply non- degenerate perturbation theory by letting the perturbation be: A, = ß2³. а) Calculate to first order the ground state energy of this anharmonic oscillator. b) ( | Calculate to second order the ground state energy of this anharmonic oscillator. c) ( Find the lowest order correction to the ground state wave function.

Consider a system spin-1/2 system, denoted by A, interacting with another system spin-1/2 system, denoted by B, such that the state of the combined system is AB) a++ B|-+). Find (a) the density matrix PA for system A corresponding to this state and (b) obtain the formulas for (()).