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# 342-PS3sol - Solutions to Problem Set 3 Physics 342 by...

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Solutions to Problem Set 3 Physics 342 by: Callum Quigley 1 Clebsh-Gordan Coefficients Let’s write | j, m i for the eigenstates of total J 2 and J z , while we’ll write | j 1 , m 1 ; j 2 , m 2 i for the tensor product representation. For this question, the rules to find CG coefficients are spelled out pretty clearly on p. 412 of Shankar. Let me simply summarize the results here. (i) 1 1 2 = 3 2 1 2 | 3 / 2 , +3 / 2 i = | 1 , +1 ; 1 / 2 , +1 / 2 i (1.1) | 3 / 2 , +1 / 2 i = p 2 / 3 | 1 , 0 ; 1 / 2 , +1 / 2 i + p 1 / 3 | 1 , +1 ; 1 / 2 , - 1 / 2 i (1.2) | 3 / 2 , - 1 / 2 i = p 2 / 3 | 1 , 0 ; 1 / 2 , - 1 / 2 i + p 1 / 3 | 1 , - 1 ; 1 / 2 , 1 / 2 i (1.3) | 3 / 2 , - 3 / 2 i = | 1 , - 1 ; 1 / 2 , - 1 / 2 i (1.4) | 1 / 2 , +1 / 2 i = p 2 / 3 | 1 , +1 ; 1 / 2 , - 1 / 2 i - p 1 / 3 | 1 , - 1 ; 1 / 2 , +1 / 2 i (1.5) | 1 / 2 , - 1 / 2 i = p 1 / 3 | 1 , 0 ; 1 / 2 , - 1 / 2 i - p 2 / 3 | 1 , - 1 ; 1 / 2 , +1 / 2 i (1.6) Perhaps a couple examples are in order for this calculation. The first state should be clear. To obtain the second, we have | 3 / 2 , +1 / 2 i = 1 ~ 3 J - | 3 / 2 , +3 / 2 i = 1 ~ 3 ( J 1 - + J 2 - ) | 1 , +1; 1 / 2 , +1 / 2 i (1.7) = p 2 / 3 | 1 , 0 ; 1 / 2 , +1 / 2 i + p 1 / 3 | 1 , +1 ; 1 / 2 , - 1 / 2 i Also, to begin the second tower (with j = 1 2 ), we take the most general form of | 1 2 , + 1 2 i possible, namely | 1 / 2 , +1 / 2 i = α | 1 , +1 ; 1 / 2 , - 1 / 2 i + β | 1 , - 1 ; 1 / 2 , +1 / 2 i , (1.8) then impose h 3 2 , + 1 2 | 1 2 , + 1 2 i = 0 and normalize.

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342-PS3sol - Solutions to Problem Set 3 Physics 342 by...

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