hw8a - PHYS 507 Answers to Homework VIII (Fall ’05) 1....

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Unformatted text preview: PHYS 507 Answers to Homework VIII (Fall ’05) 1. (a) We have | j 1- j 2 | ≤ j ≤ j 1 + j 2 which looks like the triangle inequality, but these are the quantum numbers, not the actual lengths of our vectors. In the actual triangle, the side lengths are a = p j 1 ( j 1 + 1), b = p j 2 ( j 2 + 1) and c = p j ( j + 1) (where I have omitted ¯ h multiplier) which corresponds to the values of p J 2 1 , etc. If θ is the angle between ~ J 1 and ~ J 2 (I will use the internal angle), then we know cos θ = ( a 2 + b 2- c 2 ) / (2 ab ). Using this we can compute all the angles (Note: θ = 180 o corresponds to the case where ~ J 1 and ~ J 2 are exactly parallel and θ = 0 o corresponds to the case they are exactly anti-parallel). Our side lengths are a = √ 6, b = √ 2 and c = √ 12 , √ 6 , √ 2. j = 3 , cos θ =- 4 2 √ 12 , θ = 125 o , j = 2 , cos θ = 2 2 √ 12 , θ = 73 o , j = 1 , cos θ = 6 2 √ 12 , θ = 30 o . So, maximum j value corresponds to the case where the vectors ~ J 1 and ~ J 2 are more or less parallel. They are not exactly parallel due to the quantum uncertainties in the direction of these vectors. Similarly, minimum j value corresponds to the case where these vectors are more or less antiparallel but again not exactly....
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hw8a - PHYS 507 Answers to Homework VIII (Fall ’05) 1....

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