Can Hermitian Operators Commute if Their Commutator is Also Hermitian?

[bmb2009]

Homework Statement

[A,B] = C and operators A,B,C are all hermitian show that C=0

Homework Equations

The Attempt at a Solution

Since it is given that all operators are hermitian I know that A=A' B=B' and C=C' so i expanded it out to
AB-BA=C
A'B'-B'A'=C
(BA)' - (AB)'=C


I'm not real sure where I am supposed to go or what properties of hermitian operators I am supposed to used to show that AB=BA..any help would be appreciated thank

 

[vela]

Going back to AB-BA=C, what is C' equal to? How does it compare to what you derived so far?

 

[bmb2009]

Going back to AB-BA=C, what is C' equal to? How does it compare to what you derived so far?

So would I just treat C and C' as separate equations, equate them and show that it equals zero?

ie:

C'=AB-BA
C=(BA)'-(AB)' and just use the fact that all the operators are hermitian?

 

[vela]

Not exactly. You have C=AB-BA, so C' = (AB-BA)'. With a little algebra, you should be able to show that C' = -C.