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Matrix Elements of Vectors

  1. Jul 24, 2014 #1
    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!
     
  2. jcsd
  3. Jul 24, 2014 #2

    atyy

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    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 [Broken]
    http://web.uconn.edu/~ch351vc/pdfs/spin1.pdf

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

    To get the matrix element that is in the second row and first column of the σz Pauli matrix, one does:
    <spin in down direction|σz|spin in up diection> = [0 1]σz[1 0]T
     
    Last edited by a moderator: May 6, 2017
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