hw9 - 8-1 2 2 2 2 2 n1 n2 n 3 E 2 m Lx Ly L z 22 2 2 2 Lx L...

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8-1 E 2 2 2 m n 1 L x 2 n 2 L y 2 n 3 L z 2 L x L , L y L z 2 L . Let 2 2 8 mL 2 E 0 . Then E E 0 4 n 1 2 n 2 2 n 3 2 . Choose the quantum numbers as follows: n 1 n 2 n 3 E E 0 1 1 1 6 ground state 1 2 1 9 * first two excited states 1 1 2 9 * 2 1 1 18 1 2 2 12 * next excited state 2 1 2 21 2 2 1 21 2 2 2 24 1 1 3 14 * next two excited states 1 3 1 14 * Therefore the first 6 states are 111 , 121 , 112 , 122 , 113 , and 131 with relative energies E E 0 6 , 9, 9, 12, 14, 14. First and third excited states are doubly degenerate. 8-2 (a) n 1 1 , n 2 1 , n 3 1 E 0 3 2 2 2 mL 2 3 h 2 8 mL 2 3 6.626 10 34 Js 2 89 .11 10 31 kg  2 10 10 m 2 4.52 10 18 J 28.2 eV (b) n 1 2 , n 2 1 , n 3 1 or n 1 1 , n 2 2 , n 3 1 or n 1 1 , n 2 1 , n 3 2 E 1 6 h 2 8 mL 2 2 E 0 56.4 eV 8-3 n 2 11 (a) E 2 2 2 mL 2 n 2 11 2 2 2 mL 2 (b) n 1 n 2 n 3 1 1 3 1 3 1 3-fold degenerate 3 1 1 (c) 113 A sin x L sin y L sin 3 z L
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131 A sin x L sin 3 y L sin z L 311 A sin 3 x L sin y L sin z L 8-4 (a) x , y 1 x  2 y . In the two-dimensional case, A sin k 1 x sin k 2 y where k 1 n 1 L and k 2 n 2 L . (b) E 2 2 n 1 2 n 2 2 2 mL 2 If we let E 0 2 2 mL 2 , then the energy levels are: n 1 n 2 E E 0 1 1 1 11 1 2 5 2 12 doubly degenerate 2 1 5 2 21 2 2 4 22 8-5 (a) n 1 n 2 n 3 1 and E 111 3 h 2 8 mL 2 36 .63 10 34 2 81 .67 10 27 4 10 28 2.47 10 13 J 1.54 MeV (b) States 211, 121, 112 have the same energy and E 2 2 1 2 1 2 h 2 8 mL 2 2 E 111 3.08 MeV and states 221, 122, 212 have the energy E 2 2 2 2 1 2 h 2 8 mL 2 3 E 111 4.63 MeV .
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hw9 - 8-1 2 2 2 2 2 n1 n2 n 3 E 2 m Lx Ly L z 22 2 2 2 Lx L...

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