If two operators commute (eigenvector question)

In summary, the conversation discusses the properties of \Omega_1 and \Omega_2, which satisfy [\Omega_1,\Omega_2]=0 and \Omega = \Omega_1 + \Omega_2. It is mentioned that if \Psi_1 and \Psi_2 are eigenvectors of \Omega_1 and \Omega_2, respectively, then their tensor product \Psi = \Psi_1 \Psi_2 is also an eigenvector of \Omega. It is also noted that if the \Psi_i are normalized, then \Psi is automatically normalized. The conversation ends with a question about whether \Omega_i are Hermitean, but it is mentioned that this might not make a difference
  • #1
AxiomOfChoice
533
1
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?
 
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  • #2
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.)
 
  • #3
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.
 

Related to If two operators commute (eigenvector question)

What does it mean for two operators to commute?

Two operators are said to commute if they can be applied in any order and still result in the same outcome. In mathematical terms, this means that the order of multiplication of the two operators does not matter.

What is an eigenvector?

An eigenvector is a vector that, when operated on by a linear transformation or operator, results in a scalar multiple of the original vector. This scalar multiple is known as the eigenvalue of the eigenvector.

Do all operators commute?

No, not all operators commute. Only certain pairs of operators will commute, while others will not. The commutativity of operators is dependent on the specific properties and operations of the operators.

How can you tell if two operators commute?

To determine if two operators commute, you can use the commutator operation. If the commutator of the two operators is equal to zero, then they commute. If the commutator is not equal to zero, then the operators do not commute.

What is the significance of two operators commuting?

The commutativity of operators has important implications in mathematics and physics. It allows for simplified calculations and can often lead to more elegant solutions to problems. Additionally, commuting operators often correspond to observables in quantum mechanics, making them a crucial concept in understanding physical systems.

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