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Unformatted text preview: Quantum Mechanics  1: HW 6 Solutions Leo Radzihovsky Paul Martens November 12, 2007 1 Problem 1 Show that H ~ E 1 ~ E 2 commutes with ~ L , where ~ E 1 , 2 are vector operators. The definition of a vector operator. h E (1 or 2) i ,L j i = ~ ijk E k (1) (transforms like a vector under rotations) h E (1) i E (2) i ,L j i = E (1) i h E (2) i ,L j i + h E (1) i ,L j i E (2) i (2) = ~ ijk E (1) i E (2) k + E (1) k E (2) i (3) = 0 (4) Because ~ E 1 ~ E 2 commute with ~ L all functions of ~ E 1 ~ E 2 commute with ~ L h H ~ E 1 ~ E 2 , ~ L i = 0 (5) 2 Problem 2 Show that L z = ~ ( x y y x ) = ~ relating coordinates x = r sin cos (6) y = r sin sin (7) z = r cos (8) and derivatives y = r y r + y + y (9) x = r x r + x + x (10) 1 r = p x 2 + y 2 + z 2 (11) z r = cos (12) calculating derivatives r x = x r = sin cos (13) r y = y r = sin sin (14) derivatives can be taken easily by making use of the chain rule and equation (12) x z r = z r 2 r x = sin x = sin x (15) solving for x . x = 1 r cos cos (16) the calculation for y is almost identical y = 1 r cos sin (17) finding derivatives y x = tan (18) taking derivatives of both sides tan x = sec 2 x (19) solving for x . x = sin r sin (20) a similar argument can be used for y y = cos r sin (21) Putting all of it together L z ~ = x y y x = r sin cos ( y r r + y + y ) r sin sin ( x r r + x + x ) (22) L z = ~ (23) 2 3 Problem 3 Suppose we have three identical bosonic particles constrained to move on a circle in an equilateral configuration and want to find the wavefunction and the energy spectrum. The Hamiltonian of this system is H = L 2 z 2 I (24) where I = 3 mr 2 (25) The eigenstates of this Hamiltonian are the eigenstates of the L z operator. L z  ` z i = ~ ` z  ` z i (26) ` z ( ) = e ` z (27) The position of each of the particles are 1 = (28) 2 = + 2 3 (29) 3 = + 4 3 (30) Because these are bosons, we must require that the full wavefunction be symmetric under permutations. For the rigid structure above this amounts to m ( + 2 3 ) = m ( ). Therefore, ` z = 3 n , where n Z . n ( ) = e 3 n (31) E n = 9 ~ 2 n 2 2 I (32) 4 Problem 4 Express Y m ` s in Cartesian coords 4.1 Y Y = 1 4 (33) 4.2 Y 1 1 Y 1 1 = r 3 8 sin e  {z } sin cos sin sin (34) = r 3 8 x y p x 2 + y 2 + z 2 !...
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This note was uploaded on 04/09/2008 for the course PHYS 502 taught by Professor Hummin during the Fall '08 term at University of Cincinnati.
 Fall '08
 hummin
 mechanics

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