Lect11 - All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics.It has survived all

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Lecture 11, p 1 “All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics. ..It has survived all tests and there is no reason to believe that there is any flaw in it….We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.” --Murray Gell-Mann
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Lecture 11, p 2 Lecture 11: Particles in Finite Potential Wells n=1 n=2 n=3 n=4 0 L U 0 I II III U(x) y (x) AlGaAs GaAs AlGaAs U(x) x
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Lecture 11, p 3 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 11, p 4 Last Time Schrodinger’s Equation (SEQ) A wave equation that describes spatial and time dependence of Y (x,t). Expresses KE +PE = E tot Second derivative extracts -k 2 from wave function. Constraints that y (x) must satisfy Existence of derivatives (implies continuity). Boundary conditions at interfaces. Infinitely deep 1D square well (“box”) Boundary conditions  y (x) = Nsin(kx), where k = n p /L. Discrete energy spectrum: E n = n 2 E 1 , where E 1 = h 2 /8mL 2 . Normalization: N = (2/L).
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Lecture 11, p 5 Today “Normalizing” the wave function General properties of bound-state wave functions Particle in a finite square well potential Solving boundary conditions Comparison with infinite-well potential Midterm material ends here.
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Lecture 11, p 6 Particle in Infinite Square Well Potential U = U = 0 x L E n n=1 n=2 n=3 The discrete E n are known as “ energy eigenvalues ”:   2 2 2 22 2 2 11 2 1.505 where 8 n nn n p h eV nm E mm h E E n E mL electron y (x) 0 L n=1 n=2 x n=3   2 ( ) sin sin sin for 0 n n x k x x x x L L pp y    yy 2 2 2 () ( ) ( ) ( ) 2 n n n n dx U x x E x m dx  n 2L n
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Lecture 11, p 7 Probabilities n=1 |y| 2 0 x L n=2 |y| 2 0 x L |y| 2 0 x L n=3 y U= U= 0 x L y 0 x L y 0 x L Often what we measure in an experiment is the probability density, | y (x)| 2 . 1 ( ) sin n n x B x L p y    Wavefunction = Probability amplitude 2 22 1 ( ) sin n n x B x L Probability per unit length (in 1-dimension)
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Lecture 11, p 8 Probability and Normalization 1 ( ) sin n n x B x L p y    We now know that . How can we determine B 1 ?
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This note was uploaded on 04/04/2011 for the course PHYSICS 214 taught by Professor Mestre during the Spring '11 term at University of Illinois at Urbana–Champaign.

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Lect11 - All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics.It has survived all

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