Compatibility Thm HW: Can We Find More Orthon Eigenstates?

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In summary, the conversation was about the Compatibility theorem and the conditions for a set of operators to form a complete set of commuting observables (CSCO). The confusion arose from the assertion that certain eigenstates were the only ones in a given plane, and the question was whether more orthonormal eigenstates could be found. It was clarified that in the case discussed, the two eigenvectors with respect to one of the operators were unique, but if both operators had the same eigenvalue, there could be more eigenstates.
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davon806
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Homework Statement


Please see the following,I am confused by the word "only".
Q.jpg

Homework Equations

The Attempt at a Solution


I understand that the Compatibility theorem ensures we can find a basis of common eigenfunctions of [itex]\hat{A} ,\hat{B}[/itex].If each pair of eigenvalues {A_i,B_j} identifies uniquely one vector of the basis,then the set {[itex]\hat{A} ,\hat{B}[/itex]} forms a CSCO.
Here they are asserting [itex] \tilde{u_1} , \tilde{u_2} [/itex] are the only eigenstates of B in the plane.I don't see the reason for that.Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?
 
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davon806 said:
Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?
They are not degenerate with respect to B. That's exactly the case (1) discussed there, the two eigenvectors with respect to B are unique. If both eigenvalues of B are the same, see case (2).
 
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FAQ: Compatibility Thm HW: Can We Find More Orthon Eigenstates?

1. What is the Compatibility Theorem in relation to finding orthonormal eigenstates?

The Compatibility Theorem states that if two operators commute with each other, then their eigenvectors are also compatible. This means that if two operators have the same set of eigenvectors, they can be simultaneously diagonalized and the resulting eigenvectors will be orthonormal.

2. How does the Compatibility Theorem apply to quantum mechanics?

In quantum mechanics, operators represent physical observables and their eigenvectors represent the possible outcomes of measurements. The Compatibility Theorem allows us to find a set of compatible observables, whose corresponding eigenvectors can be used to fully describe the state of a quantum system.

3. Can the Compatibility Theorem be extended to more than two operators?

Yes, the Compatibility Theorem can be extended to any number of operators. If a set of operators all commute with each other, their corresponding eigenvectors will also be compatible and can be used to simultaneously diagonalize all of the operators.

4. How can the Compatibility Theorem be used in practical applications?

The Compatibility Theorem is a powerful tool in quantum mechanics and is used in various applications such as quantum computing, quantum information theory, and quantum cryptography. It allows for the efficient representation and manipulation of quantum states.

5. Are there any limitations to the Compatibility Theorem?

One limitation of the Compatibility Theorem is that it only applies to observables that commute with each other. There are cases where operators do not commute and in these situations, the Compatibility Theorem cannot be used. Additionally, the Compatibility Theorem does not guarantee that all operators can be simultaneously diagonalized, as this depends on the specific operators and their corresponding eigenvectors.

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