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[parent] constants of the motion time dependence of the statistical distribution (Application)

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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See Also: quantum operator concept, Observables and States, wave function space


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Cross-references: work, domain, volume, quantum mechanics, energy, phase factor, system, wave, motion, mechanics, characteristic function, commutes, function, operator, Hamiltonian, scalar product, observable

This is version 2 of constants of the motion time dependence of the statistical distribution, born on 2009-07-18, modified 2010-02-14.
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Physics Classification: 03.65.Ca (Formalism)
 03.65.Ta (Foundations of quantum mechanics; measurement theory )
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