Talk:PlanetPhysics/Commutation Relations of Angular Momentum

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: commutation relations of angular momentum %%% Primary Category Code: 03.65.Ca %%% Filename: CommutationRelationsOfAngularMomentum.tex %%% Version: 2 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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As an application of the \htmladdnormallink{commutator algebra}{http://planetphysics.us/encyclopedia/CommutatorAlgebra.html} rules

\begin{equation} [A,B] = -[B,A] \end{equation}

\begin{equation} [A,BC] = [A,B]C +B[A,C] \end{equation}

let us calculate the \htmladdnormallink{commutators}{http://planetphysics.us/encyclopedia/Commutator.html} of the components of the \htmladdnormallink{angular momentum}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} of a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} $$\mathbf{l} \equiv \mathbf{r} \times \mathbf{p} $$

One has

$$[l_x,l_y] = [yp_z - zp_y, zp_x - xp_z]$$

$$[l_x,l_y] = [yp_z,zp_x] + [zp_y,xp_z]$$

$$[l_x,l_y] = y[p_z,z]p_x + p_y[z,p_z]x$$

$$[l_x,l_y] = i\hbar(xp_y-yp_x)$$

$$[l_x,l_y] = i \hbar l_z$$

The other two commutators are calculated by cyclic permutation. Thus

\begin{equation} [l_x,l_y] = i \hbar l_z \,\,\,\,\,\,\, [l_y,l_z]=i\hbar l_x \,\,\,\,\,\,\, [l_z,l_x]=i \hbar l_y \end{equation}

The three components of the angular momentum do not \htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html} 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

$$[l_z,l_x^2] = i\hbar(l_yl_x+l_xl_y)$$ $$[l_z,l_y^2] = -i\hbar(l_yl_x+l_xl_y)$$ $$[l_z,l_z^2] = 0$$

Adding term by term, we obtain

\begin{equation} [l_z, \mathbf{l}^2] = 0 \end{equation}

where the \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} \begin{equation} \mathbf{l}^2 = l_x^2 + l_y^2 +l_z^2 \end{equation}

is the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} of the length of the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} $\mathbf{l}$.

The operators $\mathbf{l}^2$ and $l_z$ commute: they can therefore be simultaneously defined with infinite preicision. The pairs $(\mathbf{l}^2,l_x)$ and $(\mathbf{l}^2,l_y)$ obviosly possess the same property.

\subsection{References}

[1] Messiah, Albert. "\htmladdnormallink{Quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html}: \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html} I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

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