Download Quantum Field Theory I - 2 Problems on Assignment 7 | PHY 6645 and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 6645 Fall 2003 – Homework 7 Due by 5 p.m. on Friday, October 31. No credit will be available for homework submitted after 5 p.m. on Monday, November 3. Answer both questions. Please write neatly and include your name on the front page of your answers. You must also clearly identify all your collaborators on this assignment. To gain maximum credit you should explain your reasoning and show all working. 1. The limit of large quantum numbers. For each of the eigenstates n of the harmonic oscillator listed below under (a)–(d), generate a single graph plotting the following four quantities (clearly labeled): (i) The quantum-mechanical probability density PQM(y) = |ψn(y)|2 at dimensionless coordinate y = x √ mω/h̄; (ii) The classical probability density PCM(y) ∝ 1/v(y), where v(y) is the speed of a classical particle having the same energy as the n’th quantum-mechanical eigen- state when that particle is located at coordinate y; (iii) The box-averaged quantum-mechanical probability density P̄QM(y, w) = 1 w ∫ y+w/2 y−w/2 PQM(y)dy. (1) Use the value of w specified below. (iv) The box-averaged classical probability density P̄CM(y, w), defined by analogy with P̄QM(y, w). Note that P̄CM(y, w) may be nonzero even if y lies outside the classical turning points, so long as y − w/2 or y + w/2 is classically accessible. The horizontal axis should extend over the range 0 ≤ y ≤ ymax, where ymax is specified below. Use the same vertical scale for all four probability densities, choosing this scale so that the graph contains all of curves (i), (iii), and (iv). [PCM(y) diverges at certain points, so curve (ii) cannot be entirely contained.] Show the scale on each axis. Generate such a graph for (a) n = 0, w = 0.75, ymax = 4; (b) n = 1, w = 0.75, ymax = 4; (c) n = 14, w = 0.75, ymax = 8; (d) n = 15, w = 0.75, ymax = 8. To answer this question, you will need to produce some sort of computer program. You may use any general-purpose programming language (e.g., Fortran or C) or a higher- level programming environment (such as Mathematica, Matlab, or Maple). You may choose to represent the wave function as a set of numerical values calculated at a suffi- ciently fine grid of y points, or as an algebraic function of y which you can manipulate symbolically. The main restrictions are that you should program your own solution to the problem, not merely copy someone else’s—on this question, collaboration should be limited to discussing computational methods and comparing end results. You should not use any predefined software for generating the harmonic oscillator wave functions ψn(y), but it is acceptable to use standard library functions for the Hermite polyno- mials.
