Compatible Operators: Same Eigenvalues?

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Discussion Overview

The discussion centers on whether compatible operators in quantum mechanics must have the same eigenvalues. Participants explore the relationship between eigenvalues and eigenvectors, particularly in the context of diagonalizability and eigenspaces.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that compatible operators have the same eigenvectors or eigenspaces, but their eigenvalues can differ.
  • One participant notes that while operators can share eigenspaces, they do not necessarily have to have identical eigenspaces, emphasizing the need for diagonalizability in the same basis.
  • Another participant highlights that one operator can have different eigenvalues corresponding to the same eigenvalue of another operator.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between eigenvalues and eigenvectors of compatible operators, indicating that there is no consensus on whether they must have the same eigenvalues.

Contextual Notes

Participants mention complexities regarding eigensubspaces and diagonalizability, which may depend on specific definitions and assumptions not fully explored in the discussion.

M. next
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should compatible operators have the same eigenvalues??
 
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They have the same eigenvectors (more precise: same eigenspaces), their eigenvalues can be different. A trivial example for any operator A is the same operator with some prefactor: 2A, 3A, ...
 
mfb said:
They have the same eigenvectors (more precise: same eigenspaces), their eigenvalues can be different.

No, they don't have to have identical eigenspaces, they have to be diagonalizable in the same basis. If you want to express that in terms of eigensubspaces it gets more complicated, because eigensubspaces can be a proper subspaces of an eigenspace of the other operator only, no identity required.
 
Ah, right, one operator can have different eigenvalues for the same eigenvalue of the other operator.
 

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