If A and B are hermitian operators and C is an antihermitian operator, then

Correct option is B

Given:

• A and B are Hermitian operators.
• C is an anti-Hermitian operator.
• Analyze the nature of the commutators [[A, B], C] and [[A, C], B].

Solution:

To solve, we use the following properties:

1. Property 1:

   • The commutator of two Hermitian operators, [A, B] = AB - BA, is anti-Hermitian.

2. Property 2:

   • The commutator of a Hermitian operator and an anti-Hermitian operator, [X, Y] = XY - YX, is Hermitian.

3. Property 3:

   • The commutator of two anti-Hermitian operators, [X, Y] = XY - YX, is anti-Hermitian.

Now, analyze the two commutators:

1. For [[A, B], C]:

   • First, [A, B] is anti-Hermitian (from Property 1, since A and B are Hermitian).
   • Then, consider [[A, B], C]. Here, [A, B] is anti-Hermitian, and C is also anti-Hermitian. From Property 3, the commutator of two anti-Hermitian operators is anti-Hermitian.
   • Thus, [[A, B], C] is anti-Hermitian.

2. For [[A, C], B]:

   • First, [A, C] is Hermitian (from Property 2, since A is Hermitian and C is anti-Hermitian).
   • Then, consider [[A, C], B]. Here, [A, C] is Hermitian, and B is Hermitian. From Property 1, the commutator of two Hermitian operators is anti-Hermitian.
   • Thus, [[A, C], B] is anti-Hermitian.

Conclusion:
Both [[A, B], C] and [[A, C], B] are anti-Hermitian.

The correct answer is: (b) [[A, B], C] and [[A, C], B] are both anti-Hermitian.