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:

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

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

Failed to parse (syntax error): {\displaystyle \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

Failed to parse (syntax error): {\displaystyle \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 > }

Failed to parse (syntax error): {\displaystyle \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 $A$:

Failed to parse (syntax error): {\displaystyle i\hbar\frac{d}{dt}<A>=<[A,H]> + i\hbar\left<\frac{\partial A}{\partial t} \right> }

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 $A$.

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

$[C,H]=0$

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 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.

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].