quantum operator concept QuantumOperatorConcept 2009-03-11 00:51:18 2009-03-11 00:51:18 Topic Observables and States % 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 Consider the function $\frac{\partial \Psi}{\partial t}$, the derivative of $\Psi$ with respect to time; one can say that the operator $\frac{\partial}{\partial t}$ acting on the function $\Psi$ yields the function $\frac{\partial \Psi}{\partial t}$. More generally, if a certain operation allows us to bring into correspondence with each function $\Psi$ of a certain function space, one and only one well-defined function $\Psi^{\prime}$ of that same space, one says the $\Psi^{\prime}$ is obtained through the action of a given \emph{operator} $A$ on the function $\Psi$, and one writes $$ \Psi^{\prime} = A \Psi. $$ By definition $A$ is a \emph{linear operator} if its action on the function $\lambda_1 \Psi_1 + \lambda_2 \Psi_2$, a linear combination with constant (complex) coefficients, of two functions of this function space, is given by $$ A\left( \lambda_1 \Psi_1 + \lambda_2 \Psi_2 \right) = \lambda_1 \left( A \Psi_1 \right ) + \lambda_2 \left ( A \Psi \right ). $$ Among the linear operators acting on the wave functions $$ \Psi \equiv \Psi(\mathbf{r},t) \equiv \Psi(x,y,z,t) $$ associated with a particle, let us mention: \begin{enumerate} \item the differential operators ${\partial} / {\partial} x$,${\partial} / {\partial} y$,${\partial} / {\partial} z$,${\partial} / {\partial} t$, such as the one which was considered above; \item the operators of the form $f(\mathbf{r},t)$ whose action consists in multiplying the function $\Psi$ by the function $f(\mathbf{r},t)$ \end{enumerate} (more to come shortly)