Understanding Vector Bases in Quantum Mechanics: Explained and Applied

[player1_1_1]

Homework Statement

I am reading Ramamurti Shankar Quantum mechanics and I have a thing where is "vector base", how can I understand this "vector base"? please tell me what it means that I can ex. solve equation in eigenvector base? what it means that somewhere is a vector base? I need this to quantum mechanics, help please:) thanks!

 

[Mark44]

I don't have the text you cited and have never read it. Could the term be basis? A basis is a collection of linearly independent vectors that span a subspace of some vector space. For example, the set S = {(1, 0), (0, 1)} is a basis for R².

 

[player1_1_1]

thanks for help, and this basis is vectors which can be used to create any possible vector in this space? what is a "eigenvector basis" or something? i saw it somewhere

 

[Mark44]

thanks for help, and this basis is vectors which can be used to create any possible vector in this space? what is a "eigenvector basis" or something? i saw it somewhere

Yes, a basis can be used to generate all possible vectors in the vector space or subspace spanned by the vectors in the basis. In the simple example in my previous post, every vector in R² is a linear combination (the sum of scalar multiples) of the vectors in the basis. E.g., (3, 5) = 3(1, 0) + 5(0, 1).

An eigenvector basis is a basis consisting of a set of linearly independent eigenvectors of some linear transformation.