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 $$  = <\Psi,A\Psi>=\int \Psi^*A\Psi d \tau $$

and one has

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

The last term of the right-hand side, $$<\partial A / \partial t>$$, 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

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

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

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

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

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

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

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.