Can Hermitian Operators Commute if Their Commutator is Also Hermitian?

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Homework Help Overview

The discussion revolves around the properties of Hermitian operators and their commutation relations, specifically focusing on the commutator [A,B] = C, where A, B, and C are all Hermitian operators. The original poster attempts to demonstrate that C must equal zero under these conditions.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the Hermitian nature of the operators and question how to manipulate the commutator to derive further properties. There is a focus on understanding the relationship between C and its Hermitian conjugate C'.

Discussion Status

The discussion is active, with participants providing guidance on algebraic manipulations and questioning the relationships between the operators. Some participants suggest examining the properties of C' in relation to C, indicating a productive exploration of the problem.

Contextual Notes

There is an emphasis on the need to use the properties of Hermitian operators, and participants are considering how to equate C and C' to further their understanding. The original poster expresses uncertainty about the next steps in their reasoning.

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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
 
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Going back to AB-BA=C, what is C' equal to? How does it compare to what you derived so far?
 
vela said:
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?
 
Not exactly. You have C=AB-BA, so C' = (AB-BA)'. With a little algebra, you should be able to show that C' = -C.
 
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