Lect08 - Labs next week NOTE You will need your Active...

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Labs next week – NOTE: You will need your “Active Directory” Login {www.ad.uiuc.edu} Note: You can save a lot of time by reading the lab ahead of time – it’s a tutorial on how to draw wavefunctions.
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Particles in Finite Potential Wells n=1 n=2 n=3 n=4 U(x) 0 L U 0 I II III U →∞ U →∞ n=0 n=1 n=2 n=3 ψ( x ) x
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Overview z Particle in a finite square well potential Solving boundary conditions Comparison with infinite-well potential z Particle in a harmonic oscillator potential Electronic states in molecular potentials Vibrational states of molecules
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Example of a microscopic potential well Example of a microscopic potential well -- -- a semiconductor “quantum well” a semiconductor “quantum well” Deposit different layers of atoms on a substrate crystal: AlGaAs GaAs AlGaAs U(x) x Al As Ga Process: Molecular Beam Epitaxy Effusion cells An electron has a lower energy in GaAs than in AlGaAs. It may be trapped in the well. “Nanoscale engineering” Quantum wells like these are used for light emitting diodes and laser diodes, such as the ones used in your cd player. (The LED was developed at UIUC by Nick Holonyak)
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Particle in a Box: Review U= U= ψ (x) 0 L x n=1 n=2 n=3 |ψ| 2 U= U= 0 x L U= U= 0 x L E n n=1 n=2 n=3 z We can trap a particle in a box z The probability distribution is |ψ| 2 z The energy is proportional to n 2 ) n / L 2 ( n mL 8 h m 2 h m 2 p E 2 2 2 2 2 2 n = = = = λ
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Particle in a Finite Well (1) z What if the walls of our “box” aren’t infinitely high? “Finite Square Well” and E < U 0 Introduces very important concept of “barrier penetration” z Need to solve SEQ in 3 regions: Region II: Same as infinite square well: U(x) = 0 The general solution to the SEQ in this region is: 0 ) ( ) ( 2 ) ( 2 2 2 = + x U E m dx x d ψ = kx cos B kx sin B ) x ( 2 1 II + = E m 2 2 k 2 = = = λ π where U(x) 0 L U 0 I II III E ψ Oscillatory solutions
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Particle in a Finite Well (2) Regions I and III: U(x) = U o The general solution to this equation is: Kx Kx I e C e C x + = 2 1 ) ( ψ Region I: This is a region that is “forbidden” to classical particles. E < U 0 U(x) 0 L U 0 I II III E In I and III , the SEQ can be written: 0 ) ( ) ( 2 2 2 = x K dx x d () E U m K = 0 2 2 = where Kx Kx III e D e D x + = 2 1 ) ( Region III: (C 1 , C 2 , D 1 , and D 2 determined from boundary conditions) 0 ) ( ) ( 2 ) ( 2 2 2 = + x U E m dx x d = ψ ψ Exponential solutions
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Particle in a Finite Well (3) Kx Kx I e C e C x + = 2 1 ) ( ψ z Important new result! For quantum particles, there is a finite probability amplitude, , for finding the particle inside a “classically-forbidden” region, i.e., inside a “barrier” U(x) 0 L U 0 I II III E
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Lecture 8, Act 1 Similarly in Region III, the wavefunction must have the form 1. As x Æ , the wavefunction must vanish (why?). What does this imply for D 1 and D 2 ? 12 () K xK x III xD e D e ψ =+ U(x) 0 L U 0 I II III E b. D 2 = 0 c. Both D 1 and D 2 are nonzero a. D 1 = 0 2. What can we say about the coefficients C 1 and C 2 for the wavefunction in region I?
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This note was uploaded on 09/15/2008 for the course PHYS 214 taught by Professor Debevec during the Spring '07 term at University of Illinois at Urbana–Champaign.

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Lect08 - Labs next week NOTE You will need your Active...

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