1/2 nTx eigenfunctions of the Hamiltonian for this system are given by pn(x) n22h2 with En=2mL2, where the quantum number n can take on the values n=1, 2, 3 a. Assuming that the particle is in an eigenstate, yn(x), calculate the probability that the particle is found somewhere in the region O sXs4. Show how this probability depends on n. b. For what value of n is there the largest probability of finding the particle in 0 c. Now assume that Y is a superposition of two eigenstates, p=an+bm, at time t=0 What is Y at time t? What energy expectation value does p have at time t and how does this relate to its value at t=0? d. For an experimental measurement which is capable of distinguishing systems in state Yn from those in Ym, what fraction of a large number of systems each described by10 eigenfunctions of the Hamiltonian for this system are given by Yn(x) = è ç æ ø ÷ 2ö L 1/2 Sin npx L , with En = n2p2-h2 2mL2 , where the quantum number n can take on the values n=1,2,3,.... a. Assuming that the particle is in an eigenstate, Yn(x), calculate the probability that the particle is found somewhere in the region 0 £ x £ L 4 . Show how this probability depends on n. b. For what value of n is there the largest probability of finding the particle in 0 £ x £ L 4 ? c. Now assume that Y is a superposition of two eigenstates, Y = aYn + bYm, at time t = 0. What is Y at time t? What energy expectation value does Y have at time t and how does this relate to its value at t = 0? d. For an experimental measurement which is capable of distinguishing systems in state Yn from those in Ym, what fraction of a large number of systems each described by