PHY4604 Fall 2007 Problem Set 3 - Quantum Mechanics, Assignments of Physics

Download PHY4604 Fall 2007 Problem Set 3 - Quantum Mechanics and more Assignments Physics in PDF only on Docsity! PHY4604 Fall 2007 Problem Set 3 Department of Physics Page 1 of 2 PHY 4604 Problem Set #3 Due Wednesday October 3, 2007 (in class) (Total Points = 130, Late homework = 50%) Reading: Griffiths Chapter 2 (sections 2.4, 2.5, and 2.6). Useful Math: )( 2 1 2 1 2/)1( 0 2 + + ∞ − Γ=∫ nnaxn adxex , where Γ(x) is the gamma function and Γ(x+1) = xΓ(x). Γ(1) = Γ(2) = 1, Γ(n) = (n-1)! if n is a positive integer, and π=Γ )( 21 . Hence, a dxe ax π=∫ +∞ ∞− − 2 and 4/4/)2/()( 2222 ababbxabxxa e a dxeedxe π== ∫∫ +∞ ∞− +− +∞ ∞− +− . Problem 1 (40 points): A free particle has a Gaussian initial wave function given by 2 )0,( axAex −=Ψ , where A and a are positive real constants. (a) (2 points) Find the value of A that normalizes this wave function such that 1)0,()0,( =ΨΨ∫ +∞ ∞− ∗ dxxx . (b) (8 points) Find ),( txΨ . (c) (8 point) Compute the probability density 2|),(|),( txtx Ψ=ρ . Express your answer in terms of the time dependent quantity 2)/2(1 mat aw h+ ≡ . Sketch ρ(x,t) (as a function of x) at t = 0, and again at some very large time t. (d) (8 points) Find <x>, <x2>, <px> and <px2> for this wave function. Express your answers in terms of the time dependent quantity 2)/2(1 mat aw h+ ≡ . (e) (6 points) Compute Δx = σx and Δpx = xp σ . Is the product ΔxΔpx consistent with the uncertainty principle? At what time t does the system come closest to the uncertainty limit and how close does it get? (f) (8 points) At t = 0 what is the momentum-space wave function, φ(px), corresponding to the wave function Ψ(x,0). Compute <px> and <px2> at t = 0 using the momentum-space wave function and compare your result with what you got in (d) using the position-space wave function. (Hint: see Griffiths problem 2.22)