Talk:PlanetPhysics/Example of Quantum Commutator Algebra

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Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}. Let $f(q)$ be a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of $q$. The three \htmladdnormallink{commutators}{http://planetphysics.us/encyclopedia/Commutator.html} of $q$ and of each of the functions $p^2 f(q)$, $pf(q)p$, and $f(q)p^2$ may all be identified (to within the factor $i \hbar$) with the derivative with respect to $p$ of these functions, but they are not the same \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}. Indeed, by repeated application of the \htmladdnormallink{commutator algebra}{http://planetphysics.us/encyclopedia/CommutatorAlgebra.html} rule

\begin{equation} [q_i,G(p_1,\dots,p_R)] = i\hbar \frac{\partial G}{\partial p_i} \end{equation}

we get

$$[q,p^2 f(q)] = 2 i \hbar p f(q)$$ $$[q,pfp] = i \hbar(fp+pf)$$ $$[q,fp^2] = 2 i \hbar f p$$

In the same way

$$[p,p^2f] = \frac{\hbar}{i} p^2 f^{\prime}$$ $$[p,pfp] = \frac{\hbar}{i} pf^{\prime}p$$ $$[p,fp^2] = \frac{\hbar}{i} f^{\prime}p^2$$

\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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