The length of the box is about 13 since the electron

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Unformatted text preview: ⎬ ⎪ ⎪ ⎭ 9 Eigenvalues and normaliza1on eigenvalue 22 2 2 d 2 ⎛ nπ x ⎞ π n ⎛ nπ x ⎞ Hψ = − c sin ⎜ = c sin ⎜ ⎝L⎟ ⎠ ⎝L⎟ ⎠ 2 m dx 2 2 m L2 2π 2 n 2 ⇒E= 2 m L2 normalization nπ x L ⎛ ⎞ y= ; dx = dy; x = L → y = nπ ⎟ ⎜ ⎝ ⎠ L nπ L ⎛ nπ x ⎞ 1= ∫ c 2 cos 2 ⎜ dx = c 2 ⎝L⎟ ⎠ nπ 0 L L1 2L =c nπ = c =1 nπ 2 2 2 ⇒c= L nπ cos 2 ( y ) dy ∫ 0 2 10 Double bond spectroscopy L It is possible to think on the π system of ethene (C2H4) as a box for the electron. The length of the box is about 1.3Å . Since the electron extends beyond the carbon we take the box to be a liale larger ~ 2Å. 11 Transi1on from ground state to first excited state √(2/L) Sin(2πx/L) 2π 2 2 E= 2 2 mL2 n=2 n=1 √(2/L) Sin(πx/L) 2π 2 2 E= 1 2 mL2 12 The excita1on energy from ground to excited state ΔE = E2 − E1 2π 2 2 2 3 2π 2 2 −1 = 2 2 mL 2 mL2 ( 3 ⋅ (10 −34 2 ⋅ 9.11 ⋅ 10 ) Js ) ⋅ ( 3.141) kg ⋅ ( 3 ⋅ 10 m ) 2 −31 2 −10 2 = 1.8 ⋅ 10 −18 J To estimate the wavelength we use ΔE = hv = hc / λ λ = hc / ΔE = 6.6 ⋅ 10 −34 Js ⋅ 2.998 ⋅ 10 8 ms −1 / 1.8 ⋅ 10 −18 J λ = 1.10 ⋅ 10 −7 m = 110 nm The experimental number is 170nm. Our predic...
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This document was uploaded on 03/04/2014 for the course CH 354L at University of Texas at Austin.

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