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Lect10 - Le cture10 TheS chrdinge Equation r Lecture 10 p 1...

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Lecture 10, p 1 Lecture10: The Schrödinger Equation

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Lecture 10, p 2 This week and last week are critical for the course: Week 3, Lectures 7-9: Week 4, Lectures 10-12: Light as Particles Schrödinger 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 10-12. 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 – It’s a tutorial on how to draw wave functions. Midterm Exam Monday, Feb. 14. It will cover lectures 1-11 and some aspects of lectures 11-12. Practice exams: Old exams are linked from the course web page. Review Sunday, Feb. 13, 3-5 PM Office hours: Feb. 13 and 14
Lecture 10, p 3 Overview Probability distributions Schrödinger’s 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= ψ (x) 0 L U= n=1 n=2 x n=3

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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 1-dimensional 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?)

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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 Schrödinger proposed an equation that described the time- and space-dependence 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 time-independent SEQ . Also simplify to 1-dimension: ψ (x,y,z) ψ (x). This special case applies when the particle has a definite total energy ( E in the equation). We’ll consider the more general case ( E has a probability distribution), and also 2D and 3D motion, later.

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Lect10 - Le cture10 TheS chrdinge Equation r Lecture 10 p 1...

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