Physics 944 Homework 3: Symmetries, Perturbation Theory, Feynman-Hellmann Theorem, Assignments of Quantum Mechanics

Download Physics 944 Homework 3: Symmetries, Perturbation Theory, Feynman-Hellmann Theorem and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 944 - Homework 3 Department of Physics Due October 8, 2009 University of New Hampshire 1 Symmetries (a) [5] Sakurai problem 4.2 (b) [7] Sakurai problem 4.4 (c) [8] Sakurai problem 4.5 For this problem, consider symmetries and first order perturbation theory. 2 Time Independent Perturbation Theory (a) [10] Continuing where we left off in class, construct the eigenenergies, En, and eigenstates, |n〉, to O(λ3) and O(λ4) in perturbation theory. (b) [10] Use perturbation theory to find the first and second order relativistic correction to the one dimensional Harmonic Oscillator. 3 Feynman-Hellmann theorem Suppose the Hamiltonian, H, for a particular quantum system, is a function of some parameter λ. The solutions to the Schrödinger’s equation then give the eigenvalues E(λ) and the eigenfunctions ψ(λ) of Hψ = Eψ. (a) [10] Proof the Feynman-Hellmann theorem, which states that: ∂En ∂λ = 〈 ψ ∣∣∣∣∂H∂λ ∣∣∣∣ψ〉 (1) Hint: You need to use perturbation theory to prove this! 1