Matrix elements. LHO.

1. Mar 22, 2014

LagrangeEuler

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
2. Mar 22, 2014

Meir Achuz

I think that is why the comma is put in.

3. Mar 22, 2014

Fredrik

Staff Emeritus
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.