Chapter4_15

# Chapter4_15 - PHY4604 R D Field Addition of Angular...

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PHY4604 R. D. Field Department of Physics Chapter4_15.doc University of Florida Addition of Angular Momentum (2) Spectrum of (J 2 ) op and (J z ) op : Want to determine the spectrum of the states > >= > + >= jm j j m jm j j J jm j j j j jm j j J op z op 2 1 2 1 2 1 2 1 2 | | ) ( | ) 1 ( | ) ( and want to construct |j 1 j 2 jm> from |j 1 m 1 j 2 m 2 > = |j 1 m 1 >|j 2 m 2 > . Note that > + >= + > >= 2 2 1 1 2 1 2 2 1 1 2 2 2 1 1 1 2 2 1 1 | ) ( | ) ( | ) ( | ) ( m j m j m m m j m j J m j m j J m j m j J op z op z op z and hence > < >= < + >= < 2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 | | ) ( | ) ( | m j m j jm j j m m j m j jm j j m m m j m j J jm j j op z which implies that m = m 1 + m 2 and we know that m 1 = -j 1 , -j 1 +1,..., j 1 and m 2 = -j 2 , -j 2 +1,..., j 2 and m = -j, -j+1,..., j . Thus, m max = (m 1 ) max + (m 2 ) max = j 1 + j 2 which implies that j max = j 1 + j 2 , and there is one state such that |j 1 ,j 2 ,j=j 1 +j 2 ,m=j 1 +j 2 > = |j 1 ,m 1 =j 1 ,j 2 ,m 2 =j 2 > = |j 1 ,m 1 =j 1 >|j 2 ,m 2 =j 2 >. This is the highest m-state of the (2j max +1) states with j=j max = j 1 +j 2 . However, there are in general two states with the next lower m value: |j 1 ,j 2 ,j,m=j 1 +j 2 -1> = |j 1 ,m 1 =j 1 -1,j 2 ,m 2 =j 2 > = |j 1
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