Question 705026

SCQMEDIUM

If AA and BB are hermitian operators and CC is an antihermitian operator, then:

(A)

[[A,B],C][[A, B], C] is hermitian and [[A,C],B][[A, C], B] is antihermitian

(B)

[[A,B],C][[A, B], C] and [[A,C],B][[A, C], B] are both antihermitian

(C)

[[A,B],C][[A, B], C] and [[A,C],B][[A, C], B] are both hermitian

(D)

[[A,B],C][[A, B], C] is antihermitian and [[A,C],B][[A, C], B] is hermitian

Detailed Solution

Given A†=AA^\dagger = A, B†=BB^\dagger = B, C†=−CC^\dagger = -C. For an operator XX, it is antihermitian if X†=−XX^\dagger = -X. Calculation: [[A,B],C]†=[C†,[A,B]†]=[−C,[B,A]]=−[C,[B,A]]=−[[A,B],C][[A, B], C]^\dagger = [C^\dagger, [A, B]^\dagger] = [-C, [B, A]] = -[C, [B, A]] = -[[A, B], C]. Similarly for [[A,C],B][[A, C], B], the adjoint operation confirms it is also antihermitian.

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