# PlanetPhysics/Constants of the Motion Time Dependence of the Statistical Distribution

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Consider the Schr\"odinger equation and the complex conjugate equation:

${\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=H\Psi ,\,\,\,\,\,\,i\hbar {\frac {\partial \Psi ^{*}}{\partial t}}=-\left(H\Psi \right)^{*}}$

If ${\displaystyle \Psi }$ is normalized to unity at the initial instant, it remains normalized at any later time. The mean value of a given observable ${\displaystyle A}$ is equal at every instant to the scalar product ${\displaystyle =<\Psi ,A\Psi >=\int \Psi ^{*}A\Psi d\tau }$

and one has

$\displaystyle \frac{d}{dt} = \left < \frac{\partial \Psi}{\partial t},A\Psi \right > + \left < \Psi,A\frac{\partial \Psi}{\partial t} \right > + \left < \Psi, \frac{\partial A}{\partial t} \Psi \right >$

The last term of the right-hand side, ${\displaystyle <\partial A/\partial t>}$, is zero if ${\displaystyle A}$ does not depend upon the time explicitly.

Taking into account the Schr\"odinger equation and the hermiticity of the Hamiltonian, one has

$\displaystyle \frac{d}{dt} = - \frac{1}{i\hbar} + \frac{1}{i\hbar}<\Psi,AH\Psi> + \left< \frac{\partial A}{\partial t} \right >$

$\displaystyle \frac{d}{dt} = \frac{1}{i\hbar} <\Psi,[A,H]\Psi> + \left < \frac{\partial A}{\partial t} \right >$

Hence we obtain the general equation giving the time-dependence of the mean value of ${\displaystyle A}$:

$\displaystyle i\hbar\frac{d}{dt}=<[A,H]> + i\hbar\left<\frac{\partial A}{\partial t} \right>$

When we replace ${\displaystyle A}$by the operator ${\displaystyle e^{i\xi A}}$, we obtain an analogous equation for the time-dependence of the characterisic function of the statistical distribution of ${\displaystyle A}$.

In particular, for any variable ${\displaystyle C}$ which \htmladdnormallink{commutes {http://planetphysics.us/encyclopedia/Commutator.html} with the Hamiltonian}

${\displaystyle [C,H]=0}$

and which does not depend explicitly upon the time , one has the result

${\displaystyle {\frac {d}{dt}}=0}$

The mean value of ${\displaystyle C}$ remains constant in time. More generally, if ${\displaystyle C}$ commutes with ${\displaystyle H}$, the function ${\displaystyle e^{i\xi C}}$ also commues with ${\displaystyle H}$, and, consequently

${\displaystyle {\frac {d}{dt}}=0}$

The characteristic function, and hence the statistical distribution of the observable ${\displaystyle C}$, remain constant in time.

By analogy with Classical Analytical mechanics, ${\displaystyle C}$ is called a constant of the motion. In particular, if at the initial instant the wave function is an eigenfunction of ${\displaystyle C}$ corresponding to a give eigenvalue ${\displaystyle c}$, this property continues to hold in the course of time. One says that ${\displaystyle c}$ is a "good quantum number". If, in particular, ${\displaystyle H}$ does not explicitly depend upon the time, and if the dynamical state of the system is represented at time ${\displaystyle t_{0}}$ by an eigenfunction common to ${\displaystyle H}$ and ${\displaystyle C}$, the wave function remains unchanged in the course of time, to within a phase factor. The energy and the variable ${\displaystyle C}$ remain well defined and constant in time.

### 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].