PHY4604 R. D. Field Department of Physics Chapter4_15.doc University of Florida Addition of Angular Momentum (2) Spectrum of (J2)opand (Jz)op:Want to determine the spectrum of the states >>=>+>=jmjjmjmjjJjmjjjjjmjjJopzop212121212||)(|)1(|)(and want to construct |j1j2jm>from |j1m1j2m2> = |j1m1>|j2m2>. Note that >+>=+>>=22112122112221112211|)(|)(|)(|)(mjmjmmmjmjJmjmjJmjmjJopzopzopzand hence ><>=<+>=<22112122112121221121||)(|)(|mjmjjmjjmmjmjjmjjmmmjmjJjmjjopzwhich implies that m = m1+ m2and we know that m1= -j1, -j1+1,..., j1and m2= -j2, -j2+1,..., j2and m = -j, -j+1,..., j. Thus,mmax= (m1)max+ (m2)max= j1+ j2which implies that jmax= j1+ j2, and there is one state such that |j1,j2,j=j1+j2,m=j1+j2> = |j1,m1=j1,j2,m2=j2> = |j1,m1=j1>|j2,m2=j2>. This is the highest m-state of the (2jmax+1) states with j=jmax= j1+j2. However, there are in general two states with the next lower m value: |j1,j2,j,m=j1+j2-1> = |j1,m1=j1-1,j2,m2=j2> = |j1
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