My book on quantum physics says that if two Hermitian operators commute then it emerges that they have common eigenfunctions.(adsbygoogle = window.adsbygoogle || []).push({});

Is that true?

If A,B hermitian commuting operators and Ψ a random wavefunction then:

[A,B]Ψ=0 => ABΨ=BAΨ

If we assume that Ψ is B`s eigenfunction:

b*AΨ=BAΨ

From this equation, how does it emerge for A to have common eigenfunctions with B?

I agree that if Ψ was A`s eigenfunction then the equation would be satisfied, but in the more general case of a random Ψ it doesnt emerge that Ψ is also A`s eigenfunction!

Another argument to support this is the following.

If those operators commute, then their uncertainties satisfy the following inequality:

ΔΑ*ΔΒ >=0

Ψ is B`s eigenfunction so: ΔB=0.

Now the inequality is satisfied for ANY ΔΑ (including zero).

So is the statement << if two hermitian operators commute then it emerges that they have common eigenfunctions. >> true or false?

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# Commuting operators => Common eigenfunctions?

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