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Mathematic Methods HW Solutions 62

Mathematic Methods HW Solutions 62 - 5.14 u = 50 ln r ln 2...

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Unformatted text preview: Chapter 13 62 5.14 u = 50 ln r ln 2 + 200 π summationdisplay odd n r n- r- n n (2 n- 2- n ) sin nθ 5.15 u = 50 parenleftbigg 1- ln r ln 2 parenrightbigg- 200 π summationdisplay odd n 1 n (2 n- 2- n ) bracketleftbigg parenleftBig r 2 parenrightBig n- parenleftBig r 2 parenrightBig- n bracketrightbigg sin nθ 6.2 The first six frequencies are ν 10 , ν 11 = 1 . 593 ν 10 , ν 12 = 2 . 135 ν 10 ν 20 = 2 . 295 ν 10 , ν 13 = 2 . 652 ν 10 , ν 21 = 2 . 917 ν 10 . 6.4 ν nm = v 2 radicalBigg parenleftbigg l a parenrightbigg 2 + parenleftBig m b parenrightBig 2 + parenleftBig n c parenrightBig 2 6.5 z = 64 l 4 π 6 summationdisplay odd m summationdisplay odd n 1 n 3 m 3 sin nπx l sin mπy l cos πv ( m 2 + n 2 ) 1 / 2 t l 6.6 Ψ n = sin n x πx l sin n y πy l e- iE n t/ ¯ h , E n = π 2 ¯ h 2 ( n x 2 + n y 2 ) 2 ml 2 6.7 See Problem 6.3. Some other examples of degeneracy: ( n x ,n y ) = (1 , 8) , (8 , 1), (4 , 7) , (7 , 4), giving E n = 65 π 2 ¯ h 2 2 ml 2 ; similarly 2 2 + 9 2...
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