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Unformatted text preview: Lecture 10, p 1 Lecture 10: The Schrdinger Equation Lecture 10, p 2 This week and last week are critical for the course: Week 3, Lectures 79: Week 4, Lectures 1012: Light as Particles Schrdinger Equation Particles as waves Particles in infinite wells, finite wells Probability Uncertainty Principle Next week: Homework 4 covers material in lecture 10 due on Thur. Feb. 17. We strongly encourage you to look at the homework before the midterm! Discussion : Covers material in lectures 1012. There will be a quiz . Lab: Go to 257 Loomis (a computer room). You can save a lot of time by reading the lab ahead of time Its a tutorial on how to draw wave functions. Midterm Exam Monday, Feb. 14. It will cover lectures 111 and some aspects of lectures 1112. Practice exams: Old exams are linked from the course web page. Review Sunday, Feb. 13, 35 PM Office hours: Feb. 13 and 14 Lecture 10, p 3 Overview Probability distributions Schrdingers Equation Particle in a Box Matter waves in an infinite square well Quantized energy levels Wave function normalization Nice descriptions in the text Chapter 40 Good web site for animations http://www.falstad.com/qm1d/ U= (x) L U= n=1 n=2 x n=3 Lecture 10, p 4 Lecture 10, p 5 Having established that matter acts qualitatively like a wave, we want to be able to make precise quantitative predictions , under given conditions. Usually the conditions are specified by giving a potential energy U(x,y,z) in which the particle is located. Examples: Electron in the coulomb potential produced by the nucleus Electron in a molecule Electron in a solid crystal Electron in a nanostructure quantum dot Proton in the nuclear potential inside the nucleus x U(x) For simplicity, consider a 1dimensional potential energy function, U(x). Matter Waves  Quantitative Classically, a particle in the lowest energy state would sit right at the bottom of the well. In QM this is not possible. (Why?) Lecture 10, p 6 Act 1: Classical probability distributions x P(x) a b c E x U(x) Ball in a box: x P(x) E Ball in a valley: x U(x) a b c HINT: Think about speed vs position. Start a classical (large) object moving in a potential well (two are shown here). At some random time later, what is the probability of finding it near position x? KE Total energy E = KE + U(x) Lecture 10, p 7 In 1926, Erwin Schrdinger proposed an equation that described the time and spacedependence of the wave function for matter waves ( i.e. , electrons, protons,...) There are two important forms for the SEQ. First we will focus on a very important special case of the SEQ, the timeindependent SEQ . Also simplify to 1dimension: (x,y,z) (x)....
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 Spring '08
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