Problem Set 6 - Introduction to Quantum Mechanics I | PHY 4604, Assignments of Physics

Download Problem Set 6 - Introduction to Quantum Mechanics I | PHY 4604 and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 6 Oct. 11, 2004 Due: Oct. 18, 2004 Reading: Griffiths Ch. 2 According to classical mechanics, a particle approaching a “potential barrier” (e.g. a local maximum Vmax in the potential V (x)) will be reflected back if its energy E is less than Vmax but will move past Vmax if E is greater than Vmax. (One consequence is that a particle inside a “square well”, like a box where V > E outside, is trapped and has no probability of being found outside.) In quantum mechanics, on the other hand, such a particle has a finite probability of penetrating the barrier, whatever its energy, and a finite probability of being reflected from it. It can “leak out” of potential wells. The probability of transmission (reflection) can be expressed in terms of the trans- mission coefficient T (reflection coefficient R ), defined as the ratio of the probability flux of the transmitted (reflected) wave to the probability flux of the incident wave. Recall a general probability current is written j = − ih̄ 2m [ψ∗∇ψ − ((∇ψ∗)ψ)] (1) and the transmission and reflection coefficients are defined: T = |jT | |jI | , R = |jR| |jI | (2) where T, R and I represent the transmitted, reflected, and incident fluxes, respec- tively. Note number conservation means |jT |+ |jR| = |jI | and therefore T + R = 1. 1