L07S07_Quantum1D 01-31-07

L07S07_Quantum1D 01-31-07 - CHEM 1A L7"Quantum...

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Unformatted text preview: CHEM 1A L7: "Quantum Mechanics in 1-Dimension" Particle DeBroglie (nm) ~ 600 ~6 ~ 0.06 ~ 6x10-26 L8-3 Summary Photon (yellow) (yellow) e(v ~ 105 m sec-1) Na (80K, v~300 m sec-1) Baseball (170g, v~40 m sec-1) A.Pines,M.Kubinec,UCB Absorption and Emission Spectra E=h E=h 2004 M.Kubinec A.Pines,M.Kubinec,UCB L07-1 L8-7 An electron as a standing wave around the nucleus 2004 M.Kubinec L07-2 Absorption and Emission Spectra Electronic Transtions v E = h 2004 M.Kubinec L07-3 2004 M. Kubinec 1 CHEM 1A L7: "Quantum Mechanics in 1-Dimension" ChemQuiz 6.3 Shown is an energy energy level scheme and the emission spectrum. Which line arises from 3 1? 3 2 1 A B C E 3) C L07-4 1) A 2004 M.Kubinec 2) B (x) = A sin 2 x known momentum () ( h de Broglie relation: mv = p = Uncertainty Principle 2(x) = intensity/probability Summary uncertainty in position i Schrdinger equation: x Quantum Mechanics - h2 d2 (x) + V(x) (x) = ET (x) V(x) 82m dx2 Kinetic Potential = 0 Total (x) = A sin(kx) + B cos(kx) sin(k cos(k (0) = 0 0 x x 2004 M.Kubinec bead on a string L L (x) = Ai sin 2 x i x p ~h L19-2 B=0 n k= L uncertainty in momentum 2004 M.Kubinec known position x 0 (L) = 0 L19-5 Probability/Normalization: n x dx = 1 A2 sin2 L L Probability Energy: Wave Functions (x) 2(x) En 16 0 - h2 d2 (x) = ET (x) 82m dx2 L n=4 A2 (sin(kx)cos(kx) + x) = 1 2 A= L 0 0 L L n h2 n A sin L x 82m L = ET (x) 2 n=3 9 2(x) dx = 1 (x) = 2004 M.Kubinec 2 L sin n x L particle must be somewhere L19-8 0 En = h2n2 8mL2 n=2 n=1 4 2004 M.Kubinec L07-6 1 2004 M.Kubinec L19-9 (x) = A sin 2 x known momentum () ( h de Broglie relation: mv = p = Uncertainty Principle 2(x) = intensity or probability x uncertainty in position (x) = Ai sin 2 x i i uncertainty in momentum 2004 M.Kubinec x p h/4 h/4 L07-7 known position x 2004 M. Kubinec 2 CHEM 1A L7: "Quantum Mechanics in 1-Dimension" ChemQuiz 7.1 Which is the result of narrowing the slit? 1) 2004 M.Kubinec 2) 3) L07-8 Schrdinger equation: Schr Quantum Mechanics - h2 d2 (x) + V(x) (x) = ET (x) V(x) 82m dx2 Kinetic Potential = 0 Total (x) = A sin(kx) + B cos(kx) sin(k cos(k (0) = 0 0 0 x x 2004 M.Kubinec bead on a string L L B=0 n k= L (L ) = 0 L07-10 ChemQuiz 7.2 What are the allowed wavelengths for a particle on ring of radius r? r 1) 2n 2004 M.Kubinec 2) 2r n 3) n 2r L07-11 2004 M. Kubinec 3 CHEM 1A L7: "Quantum Mechanics in 1-Dimension" Probability/Normalization: L Probability A2 sin2 0 n x dx = 1 L L A2 (sin(kx)cos(kx) + x) = 1 2 A= L 0 0 L L 2(x) dx = 1 (x) = 2004 M.Kubinec 2 L sin n x L particle must be somewhere L07-13 0 Wave Functions Energy: - h2 d2 (x) = ET (x) 82m dx2 n n 2 A sin L x 2m L 8 h2 = ET (x) n=4 (x) 2(x) En 16 n=3 9 En = h2n2 8mL2 n=2 n=1 4 1 2004 M.Kubinec L07-14 The electrons form interference patterns, the ripples shown here, and produce standing waves. Source: IBM Corporation 2004 M.Kubinec L07-15 2004 M. Kubinec 4 CHEM 1A L7: "Quantum Mechanics in 1-Dimension" Fe atoms on a copper surface Source: IBM Corporation L07-16 2004 M.Kubinec ChemQuiz 7.3 Which `bead' has bead' the lowest ground state energy? energy? bead on a string 0 x L 1) electron 2) positron 3) proton L07-17 2004 M.Kubinec Lecture Complete 2004 M.Kubinec L07-19 2004 M. Kubinec 5 ...
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