Problem Set 4 for PHY4604: Introduction to Quantum Mechanics - Fall 2004, Assignments of Physics

Download Problem Set 4 for PHY4604: Introduction to Quantum Mechanics - Fall 2004 and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 4 Sept. 22, 2004 Due: Sept. 29, 2004 Reading: Notes, Griffiths Chapter 1,2 1. Additive Hamiltonians. 1a) Show that if the potential energy V (r) in the Schrödinger equation Hψ = Eψ H = − h̄ 2∇2 2m + V can be written as a sum of functions of a single coordinate, V (r) = V1(x1) + V2(x2) + V3(x3), (1) then the time-independent (definite energy) Schrödinger equation can be decomposed into a set of 1D equations of the form ∂2ψi(xi) ∂2xi + 2m h̄2 [Ei − Vi(xi)]ψi = 0, i = 1, 2, 3 (2) with ψ(r) = ψ1(x1)ψ2(x2)ψ3(x3) and E = E1 + E2 + E3. 1b) Use this principle to find the energy levels (all) for the anisotropic 3D harmonic oscillator. H = − h̄ 2 2m ∇2 + m 2 (ω21x 2 + ω22y 2 + ω23z 2) (3) 1c) Find the ground state wave function for this problem by using ladder operators as in class. 1d) In the isotropic case ω1 = ω2 = ω3, what is the degeneracy of the energy levels, i.e. how many linearly independent eigensolutions correspond to each distinct eigenvalue? 2. Classical and Quantum “Probability Densities” in SHO. The first ex- cited wave function for the simple harmonic oscillator, corresponding to eigen- value E1 = 3h̄ω/2, is ψ1(x) = 2√ 2π1/2x0 x x0 e − 1 2 ( x x0 )2 , x0 = √ h̄ mω (4) 1