hws05-1 - PHYS 115B Homework 5 Solutions Spring 2008...

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Unformatted text preview: PHYS 115B Homework 5 Solutions Spring 2008 Problem 1 (a) Recall the one dimensional harmonic oscillator n ( x ) = A n H n ( x ) e- 2 x 2 / 2 , where = / . In RECTANGULAR coordinates, separation of variables gives: = n x ( x ) n y ( y ) n z ( z ) = A n x H n x ( x ) e- 2 x 2 / 2 A n y H n y ( y ) e- 2 y 2 / 2 A n z H n z ( z ) e- 2 z 2 / 2 = A n x A n y A n z H n x ( x ) H n y ( y ) H n z ( z ) e- 2 r 2 / 2 A = A n x A n y A n z if you are using Equation (2.35) of revised notes 2 Note that is normalized because n x ( x ) , n y ( y ) and n z ( z ) are normalized. We have n x = 1 and n y = n z = 0: A = A 1 A A = 1 2 3 / 4 = 1 2 2 3 / 4 H ( y ) = H ( z ) = 1 H 1 ( x ) = 2 x x = 2 3 r [ Y 1 ,- 1 ( , )- Y 1 , +1 ( , )] as we showed in Homework 2 Therefore we have: = A H 1 ( x ) H ( y ) H ( z ) e- 2 r 2 / 2 = 1 2 2 3 / 4 e- 2 r 2 / 2 2 x = 1 2 5 / 2 3 / 4 e- 2 r 2 / 2 2 x = 1 2 5 / 2 3 / 4 e- 2 r 2 / 2 2 2 3 r [ Y 1 ,- 1- Y 1 , +1 ] = 1 2 5 / 2 1 / 4 e- 2 r 2 / 2 8 3 r [ Y 1 ,- 1- Y 1 , +1 ] = 1 2 ( 1 , 1 ,- 1- 1 , 1 , +1 ) using Equation 2.30 in revised notes 2using Equation 2....
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This note was uploaded on 07/15/2008 for the course PHYS 115 taught by Professor Cleland during the Spring '08 term at UCSB.

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hws05-1 - PHYS 115B Homework 5 Solutions Spring 2008...

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