time dependence of the statistical distribution, constants of the motion

Consider the Schrödinger 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 $ \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

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

and one has

$\displaystyle \frac{d}{dt} <A> = \left < \frac{\partial \Psi}{\partial t},A\Psi...
...artial 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 Schrödinger equation and the hermiticity of the Hamiltonian, one has

$\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 >
$

$\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$ :

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

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 commutes with the Hamiltonian

$\displaystyle [C,H] = 0$

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

$\displaystyle \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

$\displaystyle \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].



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