Homework 1 with Solutions - Quantum Mechanics I | PHY 389K, Assignments of Quantum Mechanics

Download Homework 1 with Solutions - Quantum Mechanics I | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! PHYSICS 389K, SPRING 2005 HOMEWORK #1, Due Friday 02/02 Problem 8 page 61 in “SAKURAI” 1. Using the orthonormality of |+〉 and |−〉, prove [Si, Sj] = iijkhSk, {Si, Sj} = ( h̄2 2 ) δij, where Sx = h̄ 2 (|+〉〈−| + |−〉〈+|), Sy = ih̄ 2 (−|+〉〈−| + |−〉〈+|), Sz = h̄ 2 (|+〉〈+| − |−〉〈−|). Problem 9 page 61 in “SAKURAI” 2. Construct |S · n̂; +〉 such that S · n̂|S · n̂; +〉 = ( h̄ 2 ) |S · n̂; +〉 where n̂ is a unit vector. Express your answer as a linear combination of |+〉 and |−〉. Problem 10 page 62 in “SAKURAI” 3. The Hamiltonian operator for a two-state system is given by H = a(|1〉〈1| − |2〉〈2| + |2〉〈1| + |1〉〈2|), where a is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets (as linear combinations of |1〉 and |2〉).