Talk:PlanetPhysics/Examples of Constants of the Motion

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: examples of constants of the motion %%% Primary Category Code: 03.65.Ca %%% Filename: ExamplesOfConstantsOfTheMotion.tex %%% Version: 1 %%% 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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There exists an \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} which always \htmladdnormallink{commutes}{http://planetphysics.us/encyclopedia/Commutator.html} with the \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html}: the Hamiltonian itself. The \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} is therefore a constant of the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of all \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} whose Hamiltonian does not depend explicitly upon the time.

As another possible constant of the motion, let us mention \textbf{parity}. We denote under the name of parity the observable $P$ defined by

\begin{equation} P \psi(q) = \psi(-q) \end{equation}

It is easily verified that $P$ is Hermitean. Moreover, $P^2=1$ and, consequently, the only possible eigenvalues of $P$ are $+1$ and $-1$; even \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} are associated with $+1$, and odd functions with $-1$.

\emph{When the Hamiltonian is invariant under the substitution of} $-q$ \emph{for} $q$, we obviously have

$$[P,H] = 0$$

Indeed, if

$$H\left(\frac{\hbar}{i} \frac{d}{dq},q\right) = H\left(-\frac{\hbar}{i} \frac{d}{dq},-q\right) $$

one has, for any $\psi(q)$,

$$PH\psi = H\left(-\frac{\hbar}{i} \frac{d}{dq},-q\right)\psi(-q)=H\left(\frac{\hbar}{i} \frac{d}{dq},q\right)\psi(-q) = HP\psi$$

Under these conditions, if the \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} function has a definite parity at a given initial instant of time, it conserves the same parity in the course of time.

This property is easily extended to a system having an arbitrary number of dimensions; in particular, it applies to systems of \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} for which the parity \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} amounts to a \htmladdnormallink{reflection}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} in space $(\mathbf{r}_i \rightarrow -\mathbf{r}_i)$ and for which the observable parity is defined by

$$P\Psi(\mathbf{r}_1,\mathbf{r}_2.\dots)=\Psi(-\mathbf{r}_1,-\mathbf{r}_2,\dots)$$

\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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