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PC4201/4130 QUANTUM MECHANICS Tutorial 1 (Symmetry and Angular Momentum) 1. July 2000 Show that a necessary and sufficient condition that two linear operators A and B be equal to within a phrase, that is A B e i , is that u Av u Bv 2. for any ' ' ' A . Given that Let u and v . , show that A ( ) A 3. Show that the operator U representing a symmetry transformation must be unitary or ant-unitary: U 1 U U ( ) U 4. U ( ) U . or If a and b are any two vectors (or two vector operator operators that commute with each other and also ~ ~ with J ), show that ~ a J , b J i (( a b ) J ) ~ ~ ~ ~ ~ ~ ~ 5. Show that if an operator commutes with two of the components of the angular momentum operator, then it commutes also with the third component. 6. Show that if a system is in an eigenstate of J1 then the mean values of J2 and J3 with respect to this state vanish. Explain this result using the vector model of angular momentum. 7. Show that the spin matrix e e satisfies ~ 1 e2 0 Here e is a unit vector. ~ 1 0 ~