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 <br /> \Omega_1<br /> and <br /> \Omega_2<br /> refer to two different Hilbert spaces, then they automatically commute.