Probability densities and quantum-mechanical analogy

Let us consider a mixture of classical mobile particles A and B participating in the A -f- B —0 reaction occurring in a continuous neutral medium. Assume also that uncorrelated particles are created with rate p and recombine 

Before analysing (2.3.67), let us consider the preliminary simple statement of the problem. For the A + B B reaction without source (p = 0) the number of B particles is unchanged. Assuming also that particles do not 

As follows from (2.3.70), it is impossible to separate variables fj unless Da = 0, and mixed derivatives prevent finding of the exact solution. In the particular case Da = 0 an explicit solution of (2.3.70) is known. 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed 

Further development of this analogy leads to the non-Hermitian Hamiltonian problem describing the Bose particles. Proceeding in this way, the classical diffusion problem could be related to quantum theory of multiple scattering [115-118]. 

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