Solved Homework 6 - Quantum Field Theory | PHY 389K, Assignments of Quantum Mechanics

Download Solved Homework 6 - Quantum Field Theory | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K Quantum Mechanics, Homework Set 6 Solutions Matthias Ihl 03/23/2008 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/teaching.html 1 Problem 1 Let’s rewrite ψ(x, y, z) in terms of a radial part times an angular part: ψ(r, θ, φ) = N(sin θ cosφ+ sin θ cos φ+ cos θ)rer 2/α2 . (1) Now, let’s express the angular part in terms of spherical harmonics: ψ(r, θ, φ) = Nrer 2/α2 √ 2π 3 ( (−1 + i)Y1,1(θ, φ) + (1 + i)Y1,−1(θ, φ) + √ 2Y1,0(θ, φ) ) . (2) Because l = 1 for all spherical harmonics involved, P (L2 = 2~) = 1. More- over, P (m = 0) = |c1,0|2 |c1,−1|2 + |c1,0|2 + |c1,1|2 = 4π 3 4π 3 + 4π 3 + 4π 3 = 1/3. (3) 2 Problem 2 (a) In general, no, only if J2 = 0. (b) By the Schwartz inequality, 〈(∆Jx)2〉〈(∆Jy)2〉 ≥ |〈∆Jx∆Jy〉|2 = 1 4 |〈[∆Jx,∆Jy]〉|2 + 1 4 |〈{∆Jx,∆Jy}〉|2. (4) 1