time dependence of the statistical distribution, constants of the motion ConstantsOfTheMotionTimeDependenceOfTheStatisticalDistribution 2009-07-18 11:50:33 2009-07-18 11:50:33 Topic % 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 \setcounter{section}{1} 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 observable $A$ is equal at every instant to the scalar product $$ <A> = <\Psi,A\Psi>=\int \Psi^*A\Psi d \tau $$ 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 Hamiltonian, 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 operator $e^{i\xi A}$, we obtain an analogous equation for the time-dependence of the characterisic function of the statistical distribution of $A$. \emph{In particular, for any variable $C$ which commutes 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 characteristic function, and hence the statistical distribution of the observable $C$, remain constant in time. By analogy with Classical Analytical Mechanics, $C$ is called a \emph{constant of the motion}. In particular, if at the initial instant the wave 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 system 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 phase factor. The energy and the variable $C$ remain well defined and constant in time. \subsection{References} [1] Messiah, Albert. "Quantum Mechanics: Volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62. This entry is a derivative of the Public domain work [1].