A particle of mass m moves in a potential V(x)=kx²/2. a. If the particle has energy E, determine the range of x in which a classical particle can move. b. Determine the probability of finding the particle outside of the classical limits for the ground state. Assume that the ground state wavefunction is (x) and you can leave your answer in terms of a definite integral.

A particle in a 3-dimensional quadratic box with box length L has an energy given by h² E = (n+n+n). The degeneracies of the first, second, and 8mL² third level are a. e. 1, 2, 3 1, 3, 3 b. 1, 3, 1 c. 3, 3, 3 d. 1, 2, 2

2i+1 i+1 |- +> + 3 [recall, |+ -> means that particle #1 is in the |+> state (usual Z basis) and #2 is in the |-> state.] A) Show that this state is already normalized. B) Is this state separable or entangled? C) A measurement of S, is made on particle #1. What are the possible results and with what probabilities? D) A measurement of Sz is made on particle #2. What are the possible results and with what probabilities? E) Calculate the expectation value of the correlation function between these two measurements . (Don't use matrices -- use probabilities!)

Consider the wavefunction Y(x) = exp(-2a|x|). a) Normalize the above wavefunction. b) Sketch the probability density of the above wavefunction. c) What is the probability of finding the particle in the range 0 < x s 1/a ?