# hw7 - PHYS 507 Homework VII(Fall'05 Assigned Wednesday Due...

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PHYS 507 Homework VII (Fall ’05) Assigned: November 30, 2005, Wednesday. Due: December 7, 2005, Wednesday, at 5:00 pm. 1. We have said in class that for any vector operator ~ V , the commutation relations [ J i ,V j ] = i ¯ h k ² ijk V k are satisﬁed where ~ J is the angular momentum. These rela- tions basically say that D ~ V E transforms like vectors under rotations. (a) Show that if ~ V and ~ U are vector operators then ~ W = ~ V × ~ U is also a vector operator. (Hint: These relations might be useful, k ² ijk ² nmk = δ in δ jm - δ im δ jn and V n U m - V m U n = ij ( δ in δ jm - δ im δ jn ) V i U j .) (b) What is ~ J × ~ J ? 2. Consider the state | j,m i which is a common eigenstate of J 2 and J z . Let ˆ n = cos φ ˆ x + sin φ ˆ y be a unit vector which is perpendicular to the z -axis, but otherwise arbitrary. The component of angular momentum along ˆ n is obviously J n = ˆ n · ~ J . Compute the following (a) What is h J n i and h J 2 n i ? Do these values depend on the angle

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## This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.

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hw7 - PHYS 507 Homework VII(Fall'05 Assigned Wednesday Due...

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