Homework Assignment for Physics 651: Probability and Uncertainty in Quantum Mechanics, Assignments of Quantum Mechanics

Download Homework Assignment for Physics 651: Probability and Uncertainty in Quantum Mechanics and more Assignments Quantum Mechanics in PDF only on Docsity! Phys 651 Fall 2007 Homework #14 (due Wednesday, November 7, 2007) 1. (30 pts) Consider a wave packet freely moving in 1D so that the wave function at t = 0 is given by 2 0 2( ,0) exp 2 pxx A i x a ψ   = − +    , where p0 is a momentum of the particle, and A is the normalization constant. (a) What is the probability to find the particle in the region [ – ∆, ∆], where ∆ is a very small parameter ? (b) What is the uncertainty of the measurement of x in this state ? (c) Now consider the state of this system at some later time t and find ( , )x tψ and the probability density | ( , )x tψ |2. Hint: the easiest way is to expand ( ,0)xψ in terms of the momentum eigenstates and then propagate them in time. Make sure to check your function ( , )x tψ (that at t = 0 you get the initially given ( ,0)xψ ). Don’t be afraid of a very long expression you obtained in (c) – just rearrange the terms in a way that you can actually analyze the function in order to answer the following questions: (d) Did the probability to find the particle in the region [ – ∆, ∆] change ? If yes, how (a qualitative answer is fine) ? (e) Did the uncertainty of the measurement of x change ? If yes, how (a qualitative answer is fine)? 2. Reading assignment: Sakurai 2.2 and pp.97-100.