Answer Happy

Stuck on most of these questions, any help with them, even an
outline of how to answer them, would be greatly appreciated.
Part III: Long Answer Questions Instructions: You must answer all the questions in this Part. Please clearly show the working for these problems in your answer booklet. (a) Consider a quantum particle of mass m trapped in one dimension in an infinite square well of width a. At time t = 0 the state of the particle is given by: 1 |v, t = 0): (1) + Ai (2) (2) √3 where [n) are the normalized solutions to the time independent Schrödinger equa- tion with position space representation ₁ (x) = (x|n) = √sin (7x). (i) Determine the real positive constant A. (ii) Determine (H) at t = 0 with H the Hamiltonian operator. (iii) Give the wave-function as a function of time. (iv) Will (H) change with time? (explain your answer) (v) Give the probability that a measurement of the energy conducted at t = 0 will will yield (H). (vi) Calculate (p) as a function of time. (b) In the following three tasks, consider a hydrogen atom in the ground state. (i) Give the possible and m quantum numbers. (ii) Give the angular wave-function. (iii) Calculate the expectation value of the electron's position. =