What is the significance of matrix elements of vectors in quantum mechanics?

[bolbteppa]

What exactly is all this 'matrix elements of vectors' stuff that Landau is talking about?

I don't mean to ask people unfamiliar with this section to read it for me, so hopefully for someone who's read it - what's going on and where would I find a more modern discussion of this section (I'm sure this whole chapter should become easy when written in modern notation)?

Thanks!

 

[atyy]

A state in quantum mechanics is a vector. If one chooses a basis, it has a representation as a column vector.

An operator in quantum mechanics is a linear operators on the state, so again, if one chooses a basis, the operator has a representation as a matrix. The matrix elements are labelled according to the row and column of the matrix.

Here is an example. The Pauli matrices are a representation of the spin operators, written in the basis such the a spin in the +z direction is the column vector [1 0]^T, and a spin in the -z direction is the column vector [0 1]^T.
http://faculty.cua.edu/sober/611/Spin_and_Pauli_matrices.pdf
http://web.uconn.edu/~ch351vc/pdfs/spin1.pdf

To get the matrix element that is in the first row and first column of the σ_z Pauli matrix, one does:
<spin in up direction|σ_z|spin in up diection> = [1 0]σ_z[1 0]^T

To get the matrix element that is in the second row and first column of the σ_z Pauli matrix, one does:
<spin in down direction|σ_z|spin in up diection> = [0 1]σ_z[1 0]^T