Matrix Elements: LHO & Dirac Notation

[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.

 

[Meir Achuz]

I think that is why the comma is put in.

 

[Fredrik]

For matrix element $(a)_{i,j}$, $j$ represent column. Is this always the case?

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.