If two operators commute (eigenvector question) (1 Viewer)

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Suppose [itex]\Omega_1[/itex] and [itex]\Omega_2[/itex] satisfy [itex][\Omega_1,\Omega_2]=0[/itex] and [itex]\Omega = \Omega_1 + \Omega_2[/itex]. If [itex]\Psi_1[/itex] and [itex]\Psi_2[/itex] are eigenvectors of [itex]\Omega_1[/itex] and [itex]\Omega_2[/itex], respectively, don't we know that the (tensor?) product [itex]\Psi = \Psi_1 \Psi_2[/itex] is an eigenvector of [itex]\Omega[/itex]? Also, if the [itex]\Psi_i[/itex] are normalized, isn't [itex]\Psi[/itex] automatically normalized?
 
Well, I've just worked through this, and I think I've determined the following: If [itex]\Omega_i \Psi_i = a_i \Psi_i[/itex], we have [itex]\Omega \Psi = (a_1 + a_2) \Psi[/itex], regardless of whether [itex][\Omega_1,\Omega_2] = 0[/itex]. Is this true? (I'm assuming the [itex]\Omega_i[/itex] are Hermitean, but even that might not make any difference.)
 
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It seems you are mixing things. Either you are discussing tensor product or not. If you are discussing tensor product and if [itex]
\Omega_1
[/itex] and [itex]
\Omega_2
[/itex] refer to two different Hilbert spaces, then they automatically commute.
 

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