If two operators commute (eigenvector question)

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.)
 
1,444
4
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.
 

Want to reply to this thread?

"If two operators commute (eigenvector question)" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads

Top