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
, the formal solution is (special case:
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).
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, Cambridge: Cambridge University Press, 1997.
 P. Meystre and M. Sargent III, Elements of Quantum Optics,
Berlin Heidelberg: Springer Verlag, 1991.
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 M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions,
Frankfurt: Verlag Harri Deutsch, 1984.
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, Teil 2: Frankfurt: Verlag Harri Deutsch, 2004.
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Berlin: Deutscher Verlag der Wissenschaften, 1972.