In their original meaning, the Fokker–Planck equations are partial differential equations for conditional probabilities in the theory of Markov processes, a special class of stochastic processes. In the quantum theory of a damped optical oscillator, Markovian properties also lead to a quantum FPE, valid for the quasi-probability density

.

The above functions were found as follows.

In the real variables

,

, the FPE has the form

. The reduction to a one-dimensional equation of motion gives a simplified FPE, more complicated than the heat conduction equation:

. From the method of separation of variables in

and

, the formal solution is (special case:

):

, where

are Hermite polynomials of order

. The coefficients

follow from the initial condition

:

. The problem is that to obtain an analytical solution for

, either the integration or the summation cannot be carried out for many known elementary initial functions

. In the case chosen here, when that the initial condition is the normalized function

, then both operations (integration and summation) are possible; the result is

. Returning to the original two-dimensional problem is easy:

is one of three quasi-probability densities normalized for all

(this Demonstration shows two of three calculated functions).

[1] M. O. Scully and M. S. Zubairy,

* Quantum Optics*, Cambridge: Cambridge University Press, 1997.

[2] P. Meystre and M. Sargent III,

*Elements of Quantum Optics, *Berlin Heidelberg: Springer Verlag, 1991.

[3] J. Perina,

*Coherence of Light, *Dordrecht-Boston-Lancaster: D. Reidel Publishing Company, 1985.

[4] M. Abramowitz and I. A. Stegun,

*Pocketbook of Mathematical Functions,* Frankfurt: Verlag Harri Deutsch, 1984.

[5] W. I. Smirnow,

*Lehrgang der Höheren Mathematik*, Teil 2: Frankfurt: Verlag Harri Deutsch, 2004.

[6] W. S. Wladimirow,

*Gleichungen der mathematischen Physik, *Berlin: Deutscher Verlag der Wissenschaften, 1972.