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Unformatted text preview: Lecture 11, p 1 Lecture 11: Particles in Finite Potential Wells n=1 n=2 n=3 n=4 L U I I I I I I U(x) ψ (x) AlGaAs GaAs AlGaAs U(x) x Lecture 11, p 2 This week and last week are critical for the course: Week 3, Lectures 79: Week 4, Lectures 1012: 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 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 – It’s 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 11, p 3 Last Time Schrodinger’s Equation (SEQ) A wave equation that describes spatial and time dependence of Ψ (x,t). Expresses KE +PE = E tot Second derivative extracts k 2 from wave function. Constraints that ψ (x) must satisfy Existence of derivatives (implies continuity). Boundary conditions at interfaces. Infinitely deep 1D square well (“box”) Boundary conditions → ψ (x) = Nsin(kx), where k = n π /L. Discrete energy spectrum: E n = n 2 E 1 , where E 1 = h 2 /8mL 2 . Normalization: N = √ (2/L). Lecture 11, p 4 Lecture 11, p 5 Today “Normalizing” the wave function General properties of boundstate wave functions Particle in a finite square well potential Solving boundary conditions Comparison with infinitewell potential Midterm material ends here. Lecture 11, p 6 Par ticle in I nfinite Squar e Well Potential U = ∞ U = ∞ x L E n n=1 n=2 n=3 The discrete E n are known as “ energy eigenvalues ”: λ λ ⋅ = = = = ≡ 2 2 2 2 2 2 2 1 1 2 1.505 2 2 where 8 n n n n p h eV nm E m m h E E n E mL electron ψ (x) L n=1 n=2 x n=3 ( 29 2 ( ) sin sin sin for n n n n x k x x x x L L π π ψ λ ∝ = = ≤ ≤ ψ ψ ψ + = h 2 2 2 ( ) ( ) ( ) ( ) 2 n n n n d x U x x E x m dx λ = n 2L n Lecture 11, p 7 Probabilities n=1  ψ 2 x L n=2  ψ 2 x L  ψ 2 x L n=3 ψ U= ∞ U= ∞ x L ψ x L ψ x L Often what we measure in an experiment is the probability density,  ψ (x) 2 ....
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 Spring '08
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 Light, wave function, Boundary conditions, par ti cl, Fi ni te, ni te Wel

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