Are all eigenstates of observables orthogonal?

In summary, eigenstates of observables O1 and O2 can have the same eigenvalue for O1, but still be orthogonal if they have different eigenvalues for O2. This is known as degeneracy and can occur in systems with multiple quantum numbers.
  • #1
AlonsoMcLaren
90
2
Suppose psi1 and psi2 are eigenstates of observables O1 and O2

Suppose Value of O1 of psi1 = value of O1 of psi2

Therefore, <psi1|psi2>=1

Suppose value of O2 of psi1<>value of O2 of psi2

Therefore <psi1|psi2>=0

Contradiction!how to explain
 
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  • #2
AlonsoMcLaren said:
Suppose psi1 and psi2 are eigenstates of observables O1 and O2

Suppose Value of O1 of psi1 = value of O1 of psi2

Therefore, <psi1|psi2>=1

This is not correct. Eigenvalues can be degenerate; that is, there can be more than one eigenstate for a particular eigenvalue. Example: in hydrogen, the energy only depends on the quantum number n, and not on the angular-momentum quantum numbers l and m. Eigenstates with the same value of n (and hence the same energy eigenvalue), but different values of l or m, are orthogonal.
 

FAQ: Are all eigenstates of observables orthogonal?

What is orthogonality in mathematics?

Orthogonality in mathematics refers to the relationship between two vectors or sets of data that are perpendicular or at a 90 degree angle to each other. In other words, they do not share any common direction or have any correlation between them.

Why is orthogonality important in mathematics?

Orthogonality is important in mathematics because it allows us to simplify complex problems by breaking them down into smaller, independent components. It also helps in solving systems of equations and finding the best fit for data sets.

How is orthogonality used in real-world applications?

Orthogonality is used in various fields such as engineering, physics, and computer science. In engineering, it is used to determine the forces acting on a structure. In physics, it is used to calculate the components of a vector. In computer science, it is used in programming and data analysis to find patterns and relationships between variables.

What is the difference between orthogonality and perpendicularity?

Orthogonality and perpendicularity are often used interchangeably, but there is a subtle difference between the two. While both refer to a 90 degree angle, orthogonality specifically refers to the relationship between two vectors or sets of data, while perpendicularity can refer to any two intersecting lines or planes.

Can orthogonality be extended to higher dimensions?

Yes, orthogonality can be extended to higher dimensions. In fact, in higher dimensions, there can be more than one set of orthogonal vectors. For example, in three dimensions, there can be three mutually orthogonal vectors, while in four dimensions, there can be four sets of mutually orthogonal vectors.

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