Wolfram Alpha:

```
Commutators
-----------

Define the commutator as:  [A,B] = AB - BA

In quantum mechanics, two quantities that can be simultaneously determined
precisely have operators which commute. Consider position, x,  and momentum
p -> -ih∂/∂x

When dealing with differential operators, we need a dummy function, f, on
which to operate.

[x,p]f = (xp - px)f

=  x(-ih∂f/∂x) - (-ih(∂f/∂x)x

= -ihx∂f/∂x + ih∂(xf)/∂x

= -ihx∂f/∂x + ihx(∂f/∂x) +  ihf(∂x/∂x)

=  ihf

Therefore,

[x,p] = ih

What this means is that you cannot simultaneously measure distance and
momentum with precision.  This is the essence of the Heisenberg
Uncertainty principle.

Angular momentum, L:

Classically,

L = R x P  where L points along axis of rotation.

The cross product in cartesian coords is:

Lx = ypz - zpy
Ly = zpx - xpz
Lz = xpy - ypx

[Lx,Ly] = [ypz - zpy, zpx - xpz]
= -ihypx + ihxpy
= ih(xpy - ypx)
= ihLz

[Ly,Lz] = ihLx
[Lz,Lx] = ihLy

What this means is that you cannot simultaneously measure angular
momentum in more than one direction

Spin Angular momentum, S:

-     - -     -    -     - -     -
[σx,σy] = | 0  1 || 0 -i | - | 0 -i || 0  1 |
| 1  0 || i  0 |   | i  0 || 1  0 |
-     - -     -    -     - -     -
-     -    -      -
= | i  0 | - | -i  0 |
| 0 -i |   |  0  i |
-     -    -      -
-     -
= 2ih | 1  0 |
| 0 -1 |
-     -
= 2ihσz

Or we can write,

[Sx,Sy] = ihSz since S = hσ/2

In general we can sumarize the commutation relations as follows:

[xi,xj] = 0
[pi,pj] = 0
[x,px] = ih

Or in Kronecker Delta form:

[xi,pj] = ihδij
```