Homework Set 2 - Quantum Mechanics | PHY 215A, Assignments of Quantum Mechanics

Download Homework Set 2 - Quantum Mechanics | PHY 215A and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 215A, Quantum Mechanics: Homework Set #2 Due: Oct. 17, 2007 1. (10 points) Delta and Theta Consider the θ function θ(x − y) = 0, x < y, (1) = 1, x > y. Show that d dx θ(x − y) → δ(x − y), (2) i.e. this function satisfies the properties of the Dirac δ function. 2. (10 points) Consider the matrix that describes rigid rotations in 2D: Γ = ( cosθ sinθ −sinθ cosθ ) . (a) Show that Γ is unitary, and find its eigenvalues. (b) A rotation around the ẑ axis in 3D is a simple generalization of Γ. Give the corresponding matrix, and find orthonormal eigenvectors. Interpret the results as well as you can in the context of 3D rotations. 3. (10 points.) Hermitian and anti-Hermitian. (a) Divide the matrix H =   1 3 2i 1 2 0 2 2i i   (3) into its Hermitian and anti-Hermitian parts. (b) Prove that the eigenvalues of an anti-Hermitian matrix are imaginary. 4. (5 points.) Hermiticity. Given that Ω and Λ are Hermitian, are the following Hermitian? (a) ΩΛ. (b) ΩΛ + ΛΩ ≡ {Ω, Λ}. (c) ΩΛ - ΛΩ ≡ [Ω, Λ]. (d) i[Ω, Λ]. 5. (15 points.) Power of an Operator. If Γ is a real (Hermitian) linear operator and Γm|P >= 0 for a particular ket |P >, m > 1 being some positive integer, prove that Γ|P >= 0. Hint: start with m=2. ................................... The following problem will not be graded, but it as example of the type of Fourier transform that you must know how to do (without undue sweat). Solution will be provided with other solutions. 6. (not graded) Fourier transform. Calculate the three dimensional Fourier trans- form of the normalized Gaussian function G(r) = 1 (r◦ √ π)3 e−r 2/r2 ◦.