# hw433 - PHYS 507 Homework IV Assigned December 3 2003...

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PHYS 507 Homework IV Assigned: December 3, 2003, Wednesday. Due: December 12, 2003, Friday, at 5:00 pm. 1. Let | j,m i be common eigenstates of J 2 and J z where ~ J is an angular momentum operator. Calculate the following expectation values in this state. (a) h J x i , h J y i , h J 2 x i , J 2 y (b) h J x J y i and h J y J x i . Are these real? (c) Show that ( aJ x + bJ y ) 2 = ( a 2 + b 2 ) h J 2 x i where a and b are arbitrary real num- bers. 2. Let ~ V be a vector operator and ~ J be an angular momentum operator. We have said in class that any vector operator satisﬁes the following commutation relations [ J i ,V j ] = i ¯ h X k ² ijk V k . Let us deﬁne V ± = V x ± iV y . (a) What are [ J z ,V ± ] and [ J z ,V z ]? (b) Let | j,m i be a common eigenstate of J 2 and J z . Show that the states | ψ ± i = V ± | j,m i are eigenstates of J z with eigenvalues m ± 1. (Note: But | ψ ± i may not be eigenstates of J 2 . In other words, V ± works like the ladder operators, but not

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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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hw433 - PHYS 507 Homework IV Assigned December 3 2003...

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