PHY4604 Problem Set 7: Quantum Mechanics - Fall 2004, Assignments of Physics

Download PHY4604 Problem Set 7: Quantum Mechanics - Fall 2004 and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 7 Oct. 18, 2004 Due: Oct. 25, 2004 Reading: Notes, Griffiths Chapter 3 1. Maple problem: integrating Schrödinger’s equation. Download sepa- rately. 2. Drill. (a) Linearity. Given the following operators, which are linear? Why? i. ψ → x3ψ ii. ψ → x d dx ψ iii. ψ → ψ3 (b) Commutators. Calculate the commutators (L± are the SHO ladder op- erators defined in class): i. [x3, x d dx ] ii. [∇2, 1 r ] iii. [x2, d dy ] iv. [3, x2] v. [3, d dx ] (c) Complex numbers. Simplify: i. |3 + 2i|2 ii. Re √ x2 − 2ixy iii. Im eiωt/h̄ (d) Projecting out amplitudes. Determine the amplitude desired in each problem: i. Expand the sawtooth function f(x) =    x + a/2 < −a/2 < x < 0 x 0 < x < a/2 0 otherwise in terms of eigenstates of the infinite square well between (−a/2, a/2). Determine the amplitude of the 1st and 2nd excited states. 1