Quantum Physics I: Wave Functions, Uncertainty, and Ehrenfest's Theorem Exercises - Prof. , Assignments of Quantum Physics

Download Quantum Physics I: Wave Functions, Uncertainty, and Ehrenfest's Theorem Exercises - Prof. and more Assignments Quantum Physics in PDF only on Docsity! QUANTUM PHYSICS I PROBLEM SET 1 due September 17, before class A. Exercise your math muscles 1) compute ii 2) compute eiπ/2 3) find the general solution to − ~ 2 2m d2ψ(x) dx2 = Eψ(x), (1) for E > 0. 4) what is the solution to the problem above satisfying the conditions ψ(0) = 1, dψ(x) dx |x=0 = 0 ? (2) B. A first look at the Uncertainty Principle Consider a particle described at some particular instant of time by the wave function ψ(x) = Ae−ax 2 . 1) Determine A so ψ is normalized. 2) Compute 〈x〉, 〈x2〉 and σ2x = 〈(x− 〈x〉) 2〉. 3) Compute 〈p〉, 〈p2〉 and σ2p = 〈(p− 〈p〉) 2〉. 4) Show that by changing a one can make either σ2x or σ 2 p small, but not both at the same time. Compute σxσp. C. Ehrenfest’s theorem Prove that ∂ ∂t 〈p〉 = ∫ ∞ −∞ dx Ψ(x, t)∗ ( − ∂V (x) ∂x ) Ψ(x, t). (3) This result is one way to show that, under certain circumstances, macroscopic objects obey Newton’s law F = ma. Describe in words the connection of the formula above with Newton’s law.