Lecture 16 - Quantum mechanics

Lecture 16 - Quantum mechanics - Particle in a...

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Unformatted text preview: Particle in a One-dimensional Box L x V L x < < = V = V L x or x Particle located inside the box if V = 0: e- E V dx d m 8 h 2 2 2 2 = + E * dx d m 8 h 2 2 2 2 = + Translational motion (V=0) Quantum Mechanical Particle in a Box 2 2 2 8 mL h n E n = 2 (x) is probability distribution. Wave Frequency proportional to n. Particle Energy proportional to n 2 . E inversely related to length (L 2 ). Zero-point energy motion at T=0 K. x L n L sin 2 = n n = 1, 2, 3, E 1 4E 1 9E 1 16E 1 n=1 n=2 n=3 n=4 n (x) 2 (x) L V 2 2 2 2 8 ) ( mL n n h E i f = Calculating max for Polyenes For butadiene: measured max = 217 nm calculated max = 220 nm E C H 2 CH 2 ) 1 ( 8 2 max + = = N h c L m E hc e 5.8 ) 1 4 )( 10 26 . 6 . 6 ( ) / 10 3 ( ) 10 8 . 5 )( 10 1 . 9 ( 8 34 8 2 10 31 max + = s J s m m kg h 2 2 2 2 2 2 8 2 8 1 2 mL h N mL h N E + = E 3 E 2 ( ) 2 2 8 1 L m h N E e + = Electronic Spectra of Polyenes Polyenes have conjugated -electron systems (e.g. retinal or -carotene). max depends on length of polyene chain (L). H CH 3 C H 3 CH 3 CH 3 C H 3 O CH 3 CH 3 CH 3 C H 3 CH 3 CH 3 C H 3 CH 3 CH 3 CH 3 Retinal used in vision ( max = 380 nm) -Carotene in Photosynthesis ( max = 440 nm) ) 1 ( 8 2 max + = = N h c L m E hc e N=22 & L=30 N=12 & L=15 nm 1300 ) 23 ( 10 62 . 6 ) 10 3 ( ) 10 3 )( 10 11 . 9 ( 8 34 8 2 9 31 max =...
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Lecture 16 - Quantum mechanics - Particle in a...

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