Talk:PlanetPhysics/Constants of the Motion Time Dependence of the Statistical Distribution
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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: time dependence of the statistical distribution, constants of the motion %%% Primary Category Code: 03.65.Ca %%% Filename: ConstantsOfTheMotionTimeDependenceOfTheStatisticalDistribution.tex %%% Version: 2 %%% 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 Schr\"odinger equation and the complex conjugate equation:
$$ i \hbar \frac{\partial \Psi}{\partial t} = H \Psi, \,\,\,\,\,\, i\hbar \frac{\partial \Psi^*}{\partial t} = - \left(H\Psi\right)^* $$
If $\Psi$ is normalized to unity at the initial instant, it remains normalized at any later time. The mean value of a given \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} $A$ is equal at every instant to the \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} $$ <A> = <\Psi,A\Psi>=\int \Psi^*A\Psi d \tau $$
and one has
$$ \frac{d}{dt} <A> = \left < \frac{\partial \Psi}{\partial t},A\Psi \right > + \left < \Psi,A\frac{\partial \Psi}{\partial t} \right > + \left < \Psi, \frac{\partial A}{\partial t} \Psi \right > $$
The last term of the right-hand side, $<\partial A / \partial t>$, is zero if $A$ does not depend upon the time explicitly.
Taking into account the Schr\"odinger equation and the hermiticity of the \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html}, one has
$$ \frac{d}{dt}<A> = - \frac{1}{i\hbar}<H\Psi,A\Psi> + \frac{1}{i\hbar}<\Psi,AH\Psi> + \left< \frac{\partial A}{\partial t} \right > $$
$$ \frac{d}{dt}<A> = \frac{1}{i\hbar} <\Psi,[A,H]\Psi> + \left < \frac{\partial A}{\partial t} \right > $$
Hence we obtain the general equation giving the time-dependence of the mean value of $A$:
\begin{equation} i\hbar\frac{d}{dt}<A>=<[A,H]> + i\hbar\left<\frac{\partial A}{\partial t} \right> \end{equation}
When we replace $A$by the \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $e^{i\xi A}$, we obtain an analogous equation for the time-dependence of the characterisic \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of the statistical distribution of $A$.
\emph{In particular, for any variable $C$ which \htmladdnormallink{commutes}{http://planetphysics.us/encyclopedia/Commutator.html} with the Hamiltonian}
$$ [C,H] = 0$$
\emph{and which does not depend explicitly upon the time}, one has the result
$$ \frac{d}{dt} <C> = 0 $$
The mean value of $C$ remains constant in time. More generally, if $C$ commutes with $H$, the function $e^{i \xi C}$ also commues with $H$, and, consequently
$$ \frac{d}{dt} < e^{i \xi C} > = 0 $$
The \htmladdnormallink{characteristic function}{http://planetphysics.us/encyclopedia/Predicate.html}, and hence the statistical distribution of the observable $C$, remain constant in time.
By analogy with Classical Analytical \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}, $C$ is called a \emph{constant of the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}}. In particular, if at the initial instant the \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} function is an eigenfunction of $C$ corresponding to a give eigenvalue $c$, this property continues to hold in the course of time. One says that $c$ is a "good quantum number". If, in particular, $H$ does not explicitly depend upon the time, and if the dynamical state of the \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} is represented at time $t_0$ by an eigenfunction common to $H$ and $C$, the wave function remains unchanged in the course of time, to within a \htmladdnormallink{phase factor}{http://planetphysics.us/encyclopedia/PureState.html}. The \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and the variable $C$ remain well defined and constant in time.
\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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