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


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

If is normalized to unity at the initial instant, it remains normalized at any later time. The mean value of a given observable is equal at every instant to the scalar product

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, , is zero if 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 :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle i\hbar\frac{d}{dt}<A>=<[A,H]> + i\hbar\left<\frac{\partial A}{\partial t} \right> }

When we replace by the operator , we obtain an analogous equation for the time-dependence of the characterisic function of the statistical distribution of .

In particular, for any variable which \htmladdnormallink{commutes {} with the Hamiltonian}

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

The mean value of remains constant in time. More generally, if commutes with , the function also commues with , and, consequently

The characteristic function, and hence the statistical distribution of the observable , remain constant in time.

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


[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].