3 Problems on Quantum Mechanics - Homework 7 | PH 651, Assignments of Quantum Mechanics

Download 3 Problems on Quantum Mechanics - Homework 7 | PH 651 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 651 Fall 2007 Homework #7 (due Monday, October 15, 2007) 1. (10 pts) Consider a physical system whose Hamiltonian and initial state are given by 0 1 1 0 1 11 1 0 , 1 60 0 1 2 H ε ψ −       = − =       −    , where ε0 has the dimensions of energy. (a) What values will we obtain when measuring the energy and with what probabilities? (b) Calculate the expectation value of the Hamiltonian in both ways: (i) using the results of (a), i.e. eigenvalues and probabilities, and (ii) using the definition of an expectation value and given matrix H and the initial state. 2. (15 pts) Consider a system whose Hamiltonian and an operator A are given by the matrices 0 0 1 1 0 0 4 0 1 1 0 , 4 0 1 0 0 1 0 1 0 H A aε −       = − =       −    . (a) If we measure the energy, what values will we obtain? (b) Suppose that when we measure the energy, we obtain a value of 0ε− . Immediately afterwards, we measure A. What values will we obtain for A and with what probabilities ? (c) What is the expectation value of A? 3. Reading assignment: Sakurai, 1.1-1.5.