(adsbygoogle = window.adsbygoogle || []).push({}); Hermitian operator--prove product of operators is Hermitian if they commute

1. The problem statement, all variables and given/known data

If A and B are Hermitian operators, prove that their product AB is Hermitian if and only if A and B commute.

2. Relevant equations

1. A is Hermitian if, for any well-behaved functions f and g,

[tex]\int f^* \hat{A}g d\tau = \int g (\hat{A}f)^* d\tau[/tex]

2. If A and B are Hermitian, then (A + B) is Hermitian

3. The Eigenenfucntions of a Hermitian operator that correpsond to different eigenvalues are orthogonal.

4. Commuting Hermitian operators have simultaneous eigenfunctions.

5. The set of eigenfunctions of any Hermitian operator form a complete set.

3. The attempt at a solution

Well, it seems I could play around with this

[tex]\int f^* \hat{A}\hat{B}g d\tau = \int g (\hat{B}\hat{A}f)^* d\tau[/tex]

[tex]\int f^* \hat{B}\hat{A}g d\tau = \int g (\hat{A}\hat{B}f)^* d\tau[/tex]

and try and set the different halves equal, or something, but I can't seem to justify moving any terms around.

Property #4 has the appearance of relevance; however, I also do not think I can arbitrarily turn the operators into functions and still go forward with the proof.

Besides the above points and a little discussion of how operators distribute/commute, there doesn't seem to be much basis from my course knowledge to figure this out. But hopefully I am missing something simple?

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# Homework Help: Hermitian operator-prove product of operators is Hermitian if they commute

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