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
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.
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.
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.
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.
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.