Download Harmonic Oscillator - Quantum Mechanics - Exam and more Exams Quantum Mechanics in PDF only on Docsity! Quantum mechanics II Final exam 25 October, 2007, 08:30 Course code: CTH/GU Fysik FKA081/FIM400 Time: Thursday 25 oktober, 2007, 08:30-13:30 Place: V-huset Examiner: Stellan Östlund (0708-723201) Ling Bao (031-7723184) Additional material: Your course text. You may not have extensive handwritten notes in the book. If you do, you will be asked to exchange the book for one without notes. If absolutely necessary, you may also borrow BETA from each other. Notification of grades: Points and overall grade will be announced on http://fy.chalmers.se/˜ostlund/kvant; you will be asked to fill out a course questionnaire before picking up your grade. Instructions: Please answer all the problems or mark in the answer book that you are leaving it blank. Explain your reasoning to receive partial credit for partial solutions. If a problem is too hard, but you can solve a special case then you may receive partial credit for that. Check out the last page of formulas before you start! Problem 1 (16 pt) Consider the Harmonic oscillator H = 12(x 2 + p2). Compute the following quantities and show your work. • (2 pt) 〈n |x2 |m 〉 • (2 pt) 〈n |xp |m 〉 • (2 pt) 〈n |x4 |n 〉 − (〈n |x2 |n 〉 )2. Note, no off-diagonal elements in this case ! Consider a Hamiltonian H that includes position r, momentum p, angular momentum L and contains an ordinary kinetic energy term. Assume H is invariant under rotations, parity and time reversal and that the ground state | 0 〉 is nondegenerate. Explain why each of the following statements are true or false. • (1 pt) Every eigenstate is also invariant under parity and time reversal • (1 pt) The momentum operator p commutes with H. • (1 pt) The expectation value 〈 0 | r2 |0 〉 is nonzero. • (1 pt) The expectation value 〈 0 |L |0 〉 is zero. 1 • (1 pt) The expectation value 〈 0 |L · r |0 〉 must vanish. • (1 pt) The expectation value 〈 0 |x · p |0 〉 can be nonzero. • (1 pt) Every energy eigenstate can be written as a real function of position. • (1 pt) Explain which of the assumptions imply that 〈 0 |L2 |0 〉 must be zero. • (1 pt) Explain whether or not 〈 0 |x2p2 |0 〉 could in principle be zero. • (1 pt) From the information given, list what operators commute with H. Problem 2 (12pt) - Spin-orbit coupling in hydrogen1 We use the notation for the hydrogen wavefunctions |n, l,m 〉 where 1 ≤ n and 0 ≤ l < n. The exact energy of the hydrogen atom is given by 〈nlm |H0 |nlm 〉 = −e0/n2 where e0 = −13.6eV is the ground state energy of hydrogen. Recall that the electron has spin 12 . We now consider a coupling to the hydrogen atom of the form H = H0 + β L · s Define the hydrogen state n2s+1LJ to be the an eigenstate with angular momentum L, spin s. Traditionally, L = {S, P,D..} is used rather than {0, 1, 2, ..} for the angular momentum quantum number.2 • (2 pt) Explain why |nlm 〉 are no longer exact eigenstates of H and explain why each eigenstate can now be uniquely defined by n2s+1LJ . • (2 pt ) Write down the principal quantum numbers and symmetries associated with the states 12S1 2 and 22P 3 2 Also, specify the degeneracy of these states. • (2 pt ) What is the degeneracy of the n = 2 level, and into how many distinct energy levels is this energy split by the term βL · s. • (6 pt) Derive the expression 2 L·S = (J2−L2−S2) and write down an exact expression for the energy of the state n2s+1LJ and specify the degeneracy of the state in terms of the quantum numbers s, L and J . Problem 3 (12pt) - Rotationally invariant H A toy model Hamiltonian for a molecule with three spin half atoms is given by H = JB(SA · SB + SB · SC) + JACSC · SA 1Don’t quote results from the book; I expect the problem to be worked out completely. This is not a a problem of perturbation. 2The primary quantum number n is often omitted when the value of n is obvious, but we retain it in this problem. 2
