Practice Homework 3 - Quantum Mechanics - Fall 2007 | PH 651, Assignments of Quantum Mechanics

Download Practice Homework 3 - Quantum Mechanics - Fall 2007 | PH 651 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 651 Fall 2007 Homework #3 (due Friday, October 5, 2007) 1. (20 pts) Consider the two states 1 23iψ ϕ ϕ= + and 1 2 1 2 2 iχ ϕ ϕ= − + , where 1 2,ϕ ϕ form a complete and orthonormal basis. (a) Calculate , , ,ψ ψ χ ψ ψ χ χ χ and infer ψ χ ψ χ+ + . Are the scalar products χ ψ and ψ χ equal? (b) Calculate |ψ χ and |χ ψ . Are they equal? (c) Show that the states ψ and χ satisfy the triangle inequality (d) Show that the states ψ and χ satisfy the Schwarz inequality 2. (20 pts) In the following expressions, where A is an operator, specify the nature of each expression (i.e. whether it’s an operator, a bra or a ket) and then find its Hermitian conjugate: (a) Aψ ϕ ϕ (b) A ϕ ψ (c) A Aψ ϕ ϕ ψ (d) A iAϕ ψ ϕ ψ− (e) ( )A iAϕ ϕ ψ ψ+ 3. Reading assignment: Sakurai 1.2