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As an application of the commutator algebra rules
![$\displaystyle [A,B] = -[B,A]$](http://images.physicslibrary.org/cache/objects/836/l2h/img1.png "$\displaystyle [A,B] = -[B,A]$") |
(1) |
![$\displaystyle [A,BC] = [A,B]C +B[A,C]$](http://images.physicslibrary.org/cache/objects/836/l2h/img2.png "$\displaystyle [A,BC] = [A,B]C +B[A,C]$") |
(2) |
let us calculate the commutators of the components of the angular momentum of a particle
One has
The other two commutators are calculated by cyclic permutation. Thus
![$\displaystyle [l_x,l_y] = i \hbar l_z \,\,\,\,\,\,\, [l_y,l_z]=i\hbar l_x \,\,\,\,\,\,\, [l_z,l_x]=i \hbar l_y$](http://images.physicslibrary.org/cache/objects/836/l2h/img9.png "$\displaystyle [l_x,l_y] = i \hbar l_z \,\,\,\,\,\,\, [l_y,l_z]=i\hbar l_x \,\,\,\,\,\,\, [l_z,l_x]=i \hbar l_y$") |
(3) |
The three components of the angular momentum do not commute in pairs. There is no complete orthonormal set common to any two of them. In other words, two components of angular momentum cannot, in general, be defined simultaneously with infinite precision. Note that
Adding term by term, we obtain
![$\displaystyle [l_z, \mathbf{l}^2] = 0$](http://images.physicslibrary.org/cache/objects/836/l2h/img13.png "$\displaystyle [l_z, \mathbf{l}^2] = 0$") |
(4) |
where the operator
 |
(5) |
is the square of the length of the vector
 .
The operators
 and  commute: they can therefore be simultaneously defined with infinite preicision. The pairs
 and
 obviosly possess the same property.
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
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![[parent]](https://images.physicslibrary.org/images/uparrow.png)
![$\displaystyle [l_x,l_y] = [yp_z - zp_y, zp_x - xp_z]$](http://images.physicslibrary.org/cache/objects/836/l2h/img4.png "$\displaystyle [l_x,l_y] = [yp_z - zp_y, zp_x - xp_z]$")
![$\displaystyle [l_x,l_y] = [yp_z,zp_x] + [zp_y,xp_z]$](http://images.physicslibrary.org/cache/objects/836/l2h/img5.png "$\displaystyle [l_x,l_y] = [yp_z,zp_x] + [zp_y,xp_z]$")
![$\displaystyle [l_x,l_y] = y[p_z,z]p_x + p_y[z,p_z]x$](http://images.physicslibrary.org/cache/objects/836/l2h/img6.png "$\displaystyle [l_x,l_y] = y[p_z,z]p_x + p_y[z,p_z]x$")
![$\displaystyle [l_x,l_y] = i\hbar(xp_y-yp_x)$](http://images.physicslibrary.org/cache/objects/836/l2h/img7.png "$\displaystyle [l_x,l_y] = i\hbar(xp_y-yp_x)$")
![$\displaystyle [l_x,l_y] = i \hbar l_z$](http://images.physicslibrary.org/cache/objects/836/l2h/img8.png "$\displaystyle [l_x,l_y] = i \hbar l_z$")
![$\displaystyle [l_z,l_x^2] = i\hbar(l_yl_x+l_xl_y)$](http://images.physicslibrary.org/cache/objects/836/l2h/img10.png "$\displaystyle [l_z,l_x^2] = i\hbar(l_yl_x+l_xl_y)$")
![$\displaystyle [l_z,l_y^2] = -i\hbar(l_yl_x+l_xl_y)$](http://images.physicslibrary.org/cache/objects/836/l2h/img11.png "$\displaystyle [l_z,l_y^2] = -i\hbar(l_yl_x+l_xl_y)$")
![$\displaystyle [l_z,l_z^2] = 0$](http://images.physicslibrary.org/cache/objects/836/l2h/img12.png "$\displaystyle [l_z,l_z^2] = 0$")