HW8sol1 - Physics 731 Assignment #8, Solutions 1. (a) Using...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 731 Assignment #8, Solutions 1. (a) Using the result that a ( A- a ) = 0 for an operator A and its set of eigenvalues a , we have ( J - 1) J ( J + 1) = 0 , and thus J 3 = J . Another approach is to use spectral decomposition: J 3 = (1) 3 | 11 ih 11 | + (- 1) 3 | 1- 1 ih 1- 1 | = | 11 ih 11 | + | 1- 1 ih 1- 1 | = J . Note that the | 10 i state does not contribute because of its zero eigenvalue. (b) Expanding D (1) using the result from (a), one obtains D (1) = X n =0 (- i ) n J n n ! = 1 + J (- i ) + (- i ) 3 3! + ... + J 2 (- i ) 2 2! + (- i ) 4 4! + ... = 1- i sin J + (cos - 1) J 2 . (c) Choose n = y . Then J = J y h = i 2 - 1 1- 1 1 , J 2 =- 1 2 - 1- 1- 2 1- 1 , and d (1) ( ) = 1- i sin J + (cos - 1) J 2 = 1+cos 2- sin 2 1- cos 2 sin 2 cos - sin 2 1- cos 2 sin 2 1+cos 2 . 2. (a) The wavefunction is ( ~x ) = ( x + y + 3 z ) f ( r ) = rf ( r )(sin cos + sin sin + 3 cos ) , and hence ( ~x ) = r 4 3 (1- i ) 2 Y 11- (1 + i ) 2 Y 1- 1 + 3 Y 10 .....
View Full Document

Page1 / 3

HW8sol1 - Physics 731 Assignment #8, Solutions 1. (a) Using...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online