What are the matrix elements of the angular momentum operator?

[CrimsonFlash]

What are the "matrix elements" of the angular momentum operator?

Hello,

I just recently learned about angular momentum operator. So far, I liked expressing my operators in this way: http://upload.wikimedia.org/math/8/2/6/826d794e3ca9681934aea7588961cafe.png

I like it this way because it shows that these operators are also matrices. But I can't seem to figure out a possible notational for the angular momentum operator...
Any ideas?

 

[dextercioby]

How would a derivative of delta-Dirac show you that it is a matrix ?

 

[Bill_K]

I like it this way because it shows that these operators are also matrices. But I can't seem to figure out a possible notational for the angular momentum operator...

This is presumably in three dimensions, so you need to replace the derivative of the delta function by a gradient. Then, angular momentum would look something like x x ∇δ⁽³⁾(x - x')

 

[CrimsonFlash]

How would a derivative of delta-Dirac show you that it is a matrix ?

The dirac delta is actually supposed to be all the way to the left. Sorry, that's the only picture I could find.

 

[haael]

The angular momentum operator swaps directions. There are 3 distinct components of angular momentum operator in 3 dimensions and 6 components in 4 dimensions. In general, the whole angular momentum operator is an antisymmetric matrix, numbered by the dimensions, so it has as much components.

Regarding your "matrix elements" in the sense of position representation. First pick some specific component of the angular momentum operator. I.e.
$J_{xy}$ - swaps the direction x with direction y.
Now you want to calculate
$<a|J_{xy}|b>$
where $|a>$ and $|b>$ are position eigenvectors, i.e. Dirac deltas $\delta(q - a)$ and $\delta(q - b)$ where $q$ is the position parameter. (We can't use characters x and y here, since we have reserved them for direction indicators.)

As we said, what the angular momentum operator does with the position eigenvector is to swap it with some other direction, i.e. rotate by 90 degrees. So:
$J_{xy}|x> = |y>$
$J_{xy}|y> = -|x>$
Here, $|x>$, $|y>$ are the unit position eigenvectors pointing in x and y directions, respectively. I.e. they have position components: $[1, 0, 0]$ and $[0, 1, 0]$. Remember, x and y are not variables, they are just labels.

More generally, the angular momentum rotates a vector by 90 deg.
$J_{xy}|a> = R(z, 90)|a>$
Here, the symbol $R(z, 90)$ means "rotation matrix that rotates by 90 degrees over the axis pointing in the z direction". The z direction is the direction perpendicular both to x and y. It is always so in the case of rotations. In 3 dimensions you can think of it as the cross product of x and y. In higher dimensions, it is a bit more complex.

Going back to your original problem, the "matrix elements" in the position representation will be something like:
$<a|J_{xy}|b> = <a|R(z, 90)|b> = \delta^3(a - R(z, 90)b)$
Here, a and b are variables ranging over 3-dimensional position tuples. The second one is rotated by 90 degrees. I hope you understand.

Note that you have more angular momentum operators, $J_{xz}$ and $J_{yz}$ plus their linear combinations. You can construct an angular momentum operator for any direction and any angle this way. The "matrix elements" will be computed the same way in these cases, with the necessary modifications (rotation direction and angle).

 

[oomphgalore20]

Calculating the following matrix elements:
1) <1s l $\widehat{}l$_z l 1s>

2) <2p, m=1 l $\hat{}A$$^{\pm}$ l 2p, m=0).

If I could get an idea of what those mean, where to begin or could get referred to other useful resources, it would be much appreciated.