hw7 - 6 Quantum Mechanics in One Dimension 6-2 (a)...

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6 Quantum Mechanics in One Dimension 6-2 (a) Normalization requires 1 2  dx A 2 cos 2 L 4 L 4 2 x L dx A 2 2 1 cos 4 x L L 4 L 4 dx so A 2 L . (b) P 2 0 L 8 dx A 2 cos 2 0 L 8 2 x L dx 4 L 1 2 1 cos 4 x L dx 0 L 8 2 L L 8 2 L L 4 sin 4 x L 0 L 8 1 4 1 2 0.409 6-6 x  A cos kx B sin kx x  kA sin kx kB cos kx 2 x 2 k 2 A cos kx k 2 B sin kx 2 m 2 E U 2 mE 2 A cos kx B sin kx The Schrödinger equation is satisfied if 2 x 2 2 m 2 E U or k 2 A cos kx B sin kx 2 mE A cos kx B sin kx . Therefore E 2 k 2 2 m . 6-9 E n n 2 h 2 8 mL 2 , so E E 2 E 1 3 h 2 8 mL 2
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E 3  1240 eV nm c 2 8 938.28 10 6 eV c 2 10 5 nm 2 6.14 MeV hc E 1240 eV nm 6.14 10 6 eV 2.02 10 4 nm This is the gamma ray region of the electromagnetic spectrum. 6-10 E n n 2 h 2 8 mL 2 h 2 8 mL 2 6.63 10 34 Js 2 89 .11 10 31 kg 10 10 m 2 6.03 10 18 J 37.7 eV (a) E 1 37.7 eV E 2 37.7 2 2 151 eV E 3 37.7 3 2 339 eV E 4 37.7 4 2 603 eV (b) hf hc E n i E n f hc E n i E n f 1 240 eV nm E n i E n f For n i 4 , n f 1 , E n i E n f 603 eV 37.7 eV 565 eV , 2.19 nm n i 4 , n f 2 , 2.75 nm n i 4 , n f 3 , 4.70 nm n i 3 , n f 1 , 4.12 nm n i 3 , n f 2 , 6.59 nm n i 2 , n f 1 , 10.9 nm 6-12 E hc h 2 8 mL 2 2 2 1 2  and L 38 h mc 12 7.93 10 10 m 7.93 Å. 6-13 (a) Proton in a box of width L 0.200 nm 2 10 10 m E 1 h 2 8 m p L 2 6.626 10 34 J s 2 81 .67 10 27 kg 2 10 10 m 2 8.22 10 22 J 8.22 10 22 J 1.60 10 19 JeV 5.13 10 3 eV (b) Electron in the same box: E 1 h 2 8 m e L 2 6.626 10 34 J s 2 10 31 kg 2 10 10 m 2 1.506 10 18 J 9.40 eV .
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(c) The electron has a much higher energy because it is much less massive.
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This note was uploaded on 12/02/2008 for the course PHYS phys 2d taught by Professor Hirsch during the Fall '08 term at UCSD.

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hw7 - 6 Quantum Mechanics in One Dimension 6-2 (a)...

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