Download Introduction to Quantum Mechanics - Homework 1 | PHY 4604 and more Assignments Physics in PDF only on Docsity! PHY 4604 Fall 2008 – Homework 1 Due at the start of class on Friday, September 5. No credit will be available for homework submitted after the start of class on Wednesday, September 10. Answer all four 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. This assignment is primarily designed to provide practice with standard mathematical tech- niques encountered in wave mechanics. Your may find useful the following integrals: for real a and n = 0, 1, 2, 3, . . .,∫ ∞ 0 x2n exp(−x2/a2) dx = √π (2n)! n! (a 2 )2n+1 ∫ ∞ 0 dx (a2 + x2)n+1 = π(2n− 1)!! 2n+1a2n+1n! where n! = n · (n− 1)! with 0! = 1, and n!! = n · (n− 2)!! with 0!! = (−1)!! = 1. 1. We have not yet tackled the task of solving the Schrödinger wave equation for any specific potential V (x, t). This question addresses the “reverse engineering” process of working back from a wave function Ψ(x, t) to find the potential that gives rise to it. (a) Show that V (x, t) = 1 Ψ(x, t) [ i~ ∂Ψ(x, t) ∂t + ~ 2 2m ∂2Ψ(x, t) ∂x2 ] . (1) Many wave functions have isolated nodes at points x = xn (n = 1, 2, . . .) where Ψ(xn, t) = 0. The value of V (xn, t) must be evaluated by taking the limit of Eq. (1) as x→ xn. We will see in Chapter 2 that in the case of time-independent potentials V (x), quantum mechanics assigns a special role to stationary-state wave functions of the form Ψ(x, t) = ψ(x) exp(−iEt/~), where E is the particle’s energy and ψ(x) is its spatial wave function. (b) Find an expression for V (x) in terms of the spatial wave function. (c) Consider ψ(x) = Aa−1/2 exp(γx/a), where a is a (real) length scale (e.g., the size of the region within which the particle is confined), A is a real constant, and γ can be real or complex. (i) Find V (x) for this case. What is the condition on γ for V (x) to be real? (ii) Show that this wave function cannot be normalized for any choice of γ. (d) Consider ψ(x) = Aa−1/2 exp(γx2/a2), where a, A, and γ have the same interpre- tations as in part (c). (i) Find V (x) for this case. What is the condition on γ for V (x) to be real? (ii) Find the condition that γ must satisfy for the wave function to be normalizable. 2. Consider the wave function Ψ(x, t) = A exp[−(x2/2a2 + iEt/~)], where a is a length scale and E is an energy.
