(a) The Hamiltonian operator, I, of the particle in a box is given in Equation Q3.A. H= - 2m dx2 Equation Q3.8 Using the momentum operator, Equation Q3.b, show how the Hamiltonian operator is produced. p=-in dx Equation Q3.b (2 marks) (in) Show that the Hamiltonian is a linear operator. (3 marks) [in) Explain why all operators of practical interest are Hermitian operators. (3 marks) (b) () Use Equations Q3.a and Q3.c to calculate a general expression for the energy levels of the system described by "box. Whos = No sin Equation Q3.c (5 marks) (i) Sketch the first three wavefunctions for this system and their associated probability distribution functions, stating their energies and indicating on the sketches the number of nodes present. (12 marks)