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sakurai_solutions_1-1_1-4_1-8

# sakurai_solutions_1-1_1-4_1-8 - Quantum Mechanics 215A...

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Quantum Mechanics 215A Homework Solutions #1 Sam Pinansky October 8, 2003 People did ﬁne on this assignment. The only mistakes were on the second problem. The average was 18 . 1 / 20. 1. (5 Points) This can be proved simply by writing out both sides: - AC { D,B } + A { C,B } D - C { D,A } B + { C,A } DB (1) = - ACDB - ACBD + ACBD + ABCD - CDAB - CADB + CADB + ACDB (2) = ABCD - CDAB (3) = [ AB,CD ] (4) Done. 2. (10 Points) The problem asks you to use bra-ket algebra to prove these, but some of you converted them into matrix problems. This was not incorrect, but was not exactly in the spirit of the problem. (a) By the deﬁnition of the trace: tr( XY ) X a h a | XY | a i (5) where | a i is some orthonormal basis. Now we insert a complete set of states: X a h a | XY | a i = X a X a 0 h a | X | a 0 ih a 0 | Y | a i (6) = X a 0 X a h a 0 | Y | a ih a | X | a 0 i (7) = X a 0 h a 0 | Y X | a 0 i (8) = tr( Y X ) (9) where we have used the fact that h a 0 | Y | a i and h a | X | a 0 i are just numbers so commute, and the completeness of the

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sakurai_solutions_1-1_1-4_1-8 - Quantum Mechanics 215A...

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