ExampleOfQuantumCommutatorAlgebra 2010-02-14 15:16:41 2010-02-14 15:16:41 Example commutator algebra added in reference % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum system. Let $f(q)$ be a function of $q$. The three commutators 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 operators. Indeed, by repeated application of the commutator algebra 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 $$[p,p^2f] = \frac{\hbar}{i} p^2 f^{\prime}$$ $$[p,pfp] = \frac{\hbar}{i} pf^{\prime}p$$ $$[p,fp^2] = \frac{\hbar}{i} f^{\prime}p^2$$