If two operators commute (eigenvector question)

[AxiomOfChoice]

Suppose $\Omega_1$ and $\Omega_2$ satisfy $[\Omega_1,\Omega_2]=0$ and $\Omega = \Omega_1 + \Omega_2$. If $\Psi_1$ and $\Psi_2$ are eigenvectors of $\Omega_1$ and $\Omega_2$, respectively, don't we know that the (tensor?) product $\Psi = \Psi_1 \Psi_2$ is an eigenvector of $\Omega$? Also, if the $\Psi_i$ are normalized, isn't $\Psi$ automatically normalized?

 

[AxiomOfChoice]

Well, I've just worked through this, and I think I've determined the following: If $\Omega_i \Psi_i = a_i \Psi_i$, we have $\Omega \Psi = (a_1 + a_2) \Psi$, regardless of whether $[\Omega_1,\Omega_2] = 0$. Is this true? (I'm assuming the $\Omega_i$ are Hermitean, but even that might not make any difference.)

 

[arkajad]

It seems you are mixing things. Either you are discussing tensor product or not. If you are discussing tensor product and if $\Omega_1$ and $\Omega_2$ refer to two different Hilbert spaces, then they automatically commute.