3 Problem with Resolution on Quantum Mechanics I - Homework 5 | PHY 389K, Assignments of Quantum Mechanics

Download 3 Problem with Resolution on Quantum Mechanics I - Homework 5 | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K Quantum Mechanics, Homework Set 5 Solutions Matthias Ihl 03/03/2008 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/teaching.html 1 Problem 1 Consider a system whose density operator is ρ(t), evolving under the influence of a hamiltonian H(t). Show that the trace of ρ2 does not vary with time. Can the system evolve so as to be successively in a pure state and in a statistical mixture of states? Ans. Let U(t) ≡ U(t, 0) be a time-evolution operator. Then, ρ(t) ≡ ∑ i wi|αi, t〉〈αi, t| = ∑ i wiU(t)|αi, 0〉〈αi, 0|U †(t) = U(t) [ ∑ i wi|αi, 0〉〈αi, 0| ] U †(t) = U(t)ρ0U †(t) Because U is unitary, ρ(t)2 = U(t)ρ0U †(t)U(t)ρ0U †(t) = U(t)ρ2 0 U †(t) and Tr(ρ(t)2) = ∑ j 〈bj , t|ρ(t)2|bj , t〉 = ∑ j 〈bj, 0|U †(t)U(t)ρ20U †(t)U(t)|bj , 0〉 = ∑ j 〈bj , 0|ρ20|bj, 0〉 = Tr(ρ20) 1 Pure states satisfy Tr(ρ2) = 1 and mixed states satisfy Tr(ρ2) < 1. There- fore, the system cannot evolve successively in a pure state and in a mixture state. 2 Problem 2 Compute the propagator K(xf , tf ; xi, ti) for the simple harmonic oscillator. Ans. For the simple harmonic oscillator, S = m 2 ∫ (ẋ2 − ω2x2)dt, and x(t) satisfies the classical eqution ẍ+ ω2x = 0 Assume that u(t) is a classical solution of this equation, and define a new variable X ≡ u−1x. Then, S = m 2 ∫ tf ti [ u2Ẋ2 + d dt (uu̇X2) + ( u̇2 − d dt (uu̇) − ω2u2 ) X2 ] dt = m 2 ( u̇f uf x2f − u̇i ui x2i ) + m 2 ∫ f i u2Ẋ2dt (ẍ+ ω2x = 0) where ui = u(ti), uf = u(tf). Define one more new variable, a new time variable η ≡ ∫ dt u2 Then S = m 2 ( u̇f uf x2f − u̇i ui x2i ) + 1 2 ∫ ηf ηi X ′2(η)η (1) where the prime denotes the derivative with respect to η. From the relation ∆tk = ∫ ηk+1 ηk u2dη = u2(ηk)∆ηk + 1 2 du2 dη ∣ ∣ ηk ∆η2k +O(∆η 3 k)