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Matrix elements. LHO.

  1. Mar 22, 2014 #1
    For matrix element ##(a)_{i,j}##
    ##j## represent column. Is this always the case? For example in case of linear harmonic oscillator ##(\hat{a}^+)_{1,0}=\langle 1|\hat{a}^+|0 \rangle =1##. It is easier to me to see this in Dirac notation, because kets are columns and bras are rows.
     
    Last edited: Mar 22, 2014
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  3. Mar 22, 2014 #2

    Meir Achuz

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    I think that is why the comma is put in.
     
  4. Mar 22, 2014 #3

    Fredrik

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    Yes. It would be very confusing to use a different convention for different operators.

    Suppose that x and y are vectors and that A is an operator. We want the matrix equation that corresponds to y=Ax to look like [y]=[A][x]. The transpose of this equation is [x]T[A]T=[y]T. So if we would define the matrix [A] (which represents the operator A) as the transpose of the matrix we'd normally use, the matrix equation that corresponds to y=Ax would be [y]T=[x]T[A]. I suppose we could change the definitions of [x] and [y] as well. Then we'd end up with [y]=[x][A], which is slightly prettier, but still has the factors on the right in the "wrong" order.

    You might want to take a look at the FAQ post about the relationship between linear operators and matrices in the Math FAQ subforum of the General Math forum.
     
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