When is a given state an eigenstate of a given operator?

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Discussion Overview

The discussion revolves around identifying whether a given state is an eigenstate of a specified operator, focusing on theoretical aspects and the use of Dirac notation and matrix representation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks how to determine if a given state is an eigenstate of an operator.
  • Another participant notes the broadness of the question and inquires whether the focus is on experiments or theory.
  • A participant clarifies that they are referring to theory and expresses uncertainty about the category of the post. They seek to prove that a given state is not an eigenstate of an operator using Dirac notation and matrix representation.
  • There is a suggestion to start with the eigenvalue equation to show that the state is not an eigenvector of the operator, with a later reply proposing that it might be easier to assume the state is an eigenstate and derive that the eigenvalue is zero.
  • A participant expresses gratitude for the assistance received.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints regarding the approach to proving whether a state is an eigenstate, and no consensus is reached on a specific method or conclusion.

Contextual Notes

The discussion does not resolve the assumptions or steps involved in proving the eigenstate status, leaving some mathematical and conceptual aspects open-ended.

Kyle.Nemeth
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How do I know if some given state is and eigenstate of some given operator?
 
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This is an extremely broad question. Are you talking about experiments or theory?
 
I am referring to theory. I may have posted this in the wrong category as well, I'm not sure. But I'm trying to prove that some given state is not an eigenstate of some given operator. I'm doing this all in Dirac notation and matrix representation. So, should I start with the eigenvalue equation and show that the given state is not an eigenvector of the operator (Again, I know the operator and the state in matrix form)?
 
Kyle.Nemeth said:
So, should I start with the eigenvalue equation and show that the given state is not an eigenvector of the operator (Again, I know the operator and the state in matrix form)?
Yes, although it may be easier to assume that it is and then derive the result that the eigenvalue is zero.
 
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Thank you for the help Nugatory, much appreciated.
 

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