Talk:PlanetPhysics/Quantum Operator Concept

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: quantum operator concept %%% Primary Category Code: 03.65.Ca %%% Filename: QuantumOperatorConcept.tex %%% Version: 5 %%% 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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Consider the \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $\frac{\partial \Psi}{\partial t}$, the derivative of $\Psi$ with respect to time; one can say that the \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $\frac{\partial}{\partial t}$ acting on the function $\Psi$ yields the function $\frac{\partial \Psi}{\partial t}$. More generally, if a certain \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} 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 \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} functions

$$ \Psi := \Psi(\mathbf{r},t) := \Psi(x,y,z,t) $$

associated with a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html}, 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}

Starting from certain linear operators, one can form new linear operators by the following \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} operations:

\begin{enumerate}

\item multiplication of an operator $A$ by a constant $c$:

$$ (cA)\Psi := c(A\Psi) $$

\item the sum $S = A + B$ of two operators $A$ and $B$:

$$ S\Psi := A \Psi + B \Psi $$

\item the product $P=AB$ of an operator $B$ by the operator $A$:

\end{enumerate}

Note that in contrast to the sum, \emph{the product of two operators is not commutative}. Therein lies a very important difference between the algebra of linear operators and ordinary algebra.

The product $AB$ is not necessarily identical to the product $BA$; in the first case, $B$ first acts on the function $\Psi$, then $A$ acts upon the function $(B\Psi)$ to give the final result; in the second case, the roles of $A$ and $B$ are inverted. The difference $AB-BA$ of these two quantities is called the \emph{commutator} of $A$ and $B$; it is represented by the symbol $[A,B]$:

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

If this difference vanishes, one says that the two operators commute:

$$AB = BA$$

As an example of operators which do not commute, we mention the operator $f(x)$, multiplication by function $f(x)$, and the differential operator ${\partial} / {\partial x}$. Indeed we have, for any $\Psi$,

$$ \frac{\partial}{\partial x} f(x) \Psi = \frac{\partial}{\partial x} (f \Psi) = \frac{ \partial f}{\partial x} \Psi + f \frac{\partial \Psi}{\partial x} = \left ( \frac{\partial f}{\partial x} + f \frac{\partial}{\partial x} \right ) \Psi $$

In other words

\begin{equation} \left [ \frac{\partial}{\partial x},f(x) \right ] = \frac{\partial f}{\partial x} \end{equation}

and, in particular

\begin{equation} \left [ \frac{\partial}{\partial x},x \right ] = 1 \end{equation}

However, any pair of derivative operators such as ${\partial} / {\partial} x$,${\partial} / {\partial} y$,${\partial} / {\partial} z$,${\partial} / {\partial} t$, commute.

A typical example of a linear operator formed by sum and product of linear operators is the \htmladdnormallink{Laplacian}{http://planetphysics.us/encyclopedia/LaplaceOperator.html} operator

$$ \nabla^2 := \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $$

which one may consider as the \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} of the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} operator \htmladdnormallink{gradient}{http://planetphysics.us/encyclopedia/Gradient.html} $\nabla := \left( \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right )$, by itself.

\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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