Can Hermitian Operators Commute if Their Commutator is Also Hermitian?

How does that relate to C=C'?In summary, if operators A, B, and C are all hermitian and [A,B]=C, then C must equal 0. This can be shown by expanding the equation AB-BA=C and using the fact that hermitian operators have the property that C'=-C. Therefore, for the given equation to hold, C must equal its own negative, which can only be true if C=0.
  • #1
bmb2009
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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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  • #2
Going back to AB-BA=C, what is C' equal to? How does it compare to what you derived so far?
 
  • #3
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?
 
  • #4
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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Related to Can Hermitian Operators Commute if Their Commutator is Also Hermitian?

1. What is a Hermitian operator?

A Hermitian operator is a mathematical operator that has a special property known as self-adjointness. This means that the operator is equal to its own adjoint, or complex conjugate transpose. In other words, for a Hermitian operator A, A* = A. Hermitian operators are commonly used in quantum mechanics to represent physical observables such as energy, momentum, and angular momentum.

2. How do you determine if an operator is Hermitian?

An operator is Hermitian if it satisfies the condition A* = A, where A* is the adjoint of A. To determine if this condition is met, one can use the Hermiticity test, which involves taking the complex conjugate transpose of the operator and comparing it to the original operator. If they are equal, then the operator is Hermitian.

3. What is the significance of Hermitian operators in quantum mechanics?

Hermitian operators are important in quantum mechanics because they represent physical observables, such as energy and momentum, which are measured in experiments. The eigenvalues of Hermitian operators correspond to the possible outcomes of these measurements, and the corresponding eigenvectors represent the states of the system. Additionally, Hermitian operators play a crucial role in the mathematical formulation of quantum mechanics.

4. Are all physical observables represented by Hermitian operators?

No, not all physical observables are represented by Hermitian operators. Some observables, such as position and time, are represented by non-Hermitian operators. However, in order for a physical observable to be meaningful and have real-valued measurements, its corresponding operator must be Hermitian.

5. Can two Hermitian operators commute?

Yes, two Hermitian operators can commute, meaning that their corresponding operators can be multiplied in either order without changing the result. This is because Hermitian operators share a common set of eigenvectors, which allows them to commute. However, not all Hermitian operators commute with each other.

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