Lect12 - Lecture 12: Particle in 1D boxes, Simple Harmonic...

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Lecture 12, p 1 Lecture 12: Particle in 1D boxes, Simple Harmonic Oscillators U →∞ U →∞ x n=0 n=1 n=2 n=3 U(x) ψ (x)
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Lecture 12, 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 in 141 Loomis. Office hours: Feb. 13 and 14
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Lecture 12, p 3 Properties of Bound States Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound state wave functions in any 1D potential (even complicated ones). 1: The overall curvature of the wave function increases with increasing kinetic energy. 2: The lowest energy bound state always has finite kinetic energy -- called “zero-point” energy. Even the lowest energy bound state requires some wave function curvature (kinetic energy) to satisfy boundary conditions. 3: The n th wave function (eigenstate) has (n-1) zero-crossings. Larger n means larger E (and p), which means more wiggles. 4: If the potential U(x) has a center of symmetry (such as the center of the well above), the eigenstates will be, alternately, even and odd functions about that center of symmetry. ψ - = 2 2 2 2 ( ) for a sine wave 2 2 d x p m dx m ψ (x) 0 L n=1 n=2 x n=3
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Lecture 12, p 4
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The wave function below describes a quantum particle in a range x: 1. In what energy level is the particle? n = (a) 7 (b) 8 (c) 9 2. What is the approximate shape of the potential U(x) in which this particle is confined? ψ(
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Lect12 - Lecture 12: Particle in 1D boxes, Simple Harmonic...

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