Ph.D. Qualifying Exam: Quantum Mechanics Problems, Exams of Quantum Mechanics

Download Ph.D. Qualifying Exam: Quantum Mechanics Problems and more Exams Quantum Mechanics in PDF only on Docsity! Ph. D. Qualifying Exam January 2004 Quantum Mechanics Do 3 out of 4 problems Problem 1: The eigenfunction for the lowest spherically symmetric state of the electron in a hydrogen atom is given by V (r) = Ae-b' (a) Sketch the radial probability distribution for this state. (b) Find the value of r for which the radial probability is a maximum. This gives the Bohr radius ao " (c) Show that y (r) satisfies the Schrcidinger equation, and deduce the value of the Bohr radius in terms of h, m, and e . What is the ground state energy in terms of the Bohr radius? (d) Determine the normalization constantA interms of the Bohr radius. ' (e) Find the value of the expectation value of r. (f) Find the value of the expectation value of the potential energy. (g) Find the value of the expectation value of the kinetic energy. Problem 2: A quantum mechanical particle of mass rn is constrained to move in a cubic box of volume ,3. The particle moves freely within the box. (a) Calculate the pressure the particle exerts on the walls of the box when the particle is in the ground state. (b) Suppose the volume of the box is doubled suddenly by moving one wall of the box outward. What is the probability distribution of the energy of the particle after the expansion has taken place? (c) What is the expectation value of the energy after the expansion?