Correlation between Diffusion Equation and Schrödinger Equation

The well-known Schrdöinger equation is reasonably derived from the well-known diffusion equation. In the present study, the imaginary time is incorporated into the diffusion equation for understanding of the collision problem between two micro particles. It is revealed that the diffusivity corresponds to the angular momentum operator in quantum theory. The universal diffusivity expression, which is valid in an arbitrary material, will be useful for understanding of diffusion problems.


Introduction
For micro particles such as atoms or molecules in the homogeneous time and space of 1 2 3 , the macro behavior of their collective motions is presented by the well-known diffusion equation of

 
, , , t x x x The motion of a micro particle is presented by quantum mechanics and its behavior is investigated by using the Schrödinger equation of where is  2π h   h using the Plank constant ,  the state vector and  the Hamiltonian meaning the total energy in the given physical system [2].In case of a free particle, it is given by , 2 where is the particle mass and p D the momentum.In the present study, the correlation between (1) and (2) was investigated.It was found that the Schrödinger equation (2) is reasonably derived from the diffusion equa-tion (1) by means of using the imaginary time for (1).As a result, we revealed that the diffusivity in (1) corresponds to the angular momentum operator L in quantum mechanics.The obtained new diffusivity will be useful for understanding of an elementary process of diffusion [3].

Necessity of Imaginary Time
The micro particle in a solid crystal jumps instantly to the nearest lattice site through an energy barrier when it obtains an activation energy caused by the thermal fluctuation.The micro particle in a fluid collides with another one via the movement of the averaged free path and the particle jumps to a neighbor site.
For a Brownian particle of mass m, the well-known Langevin equation is where the velocity and the viscosity resistance f are d d , respectively [4].In (4), the  F t time-averaged value of external force satisfies   0 F t in a collision problem.Hereafter, we do    F t not discuss but the acceleration in a collision problem between two micro particles.In the three dimensional space

 
expressed as: Since the physical essence is still kept even if we consider the simplest collision problem of one dimensional case, we thus investigate a perfect elastic collision problem between a micro particle A and a particle B of the same kind.When the particle A moves at a velocity v 0 t  and collides at time with the particle B in the standstill state, if we can clarify the distinction between A and B after the collision, the particle A decelerates from the velocity v to the velocity zero and the particle B accelerates from the velocity zero to the velocity v between 0 .On the other hand, if we cannot clarify the distinction between A and B after the collision, it seems that the particle A decelerates from the velocity t t    v 0 to the velocity zero between t    and subsequently accelerates again from the velocity zero to the velocity v between t t . In other words, the particle motion seems as if there is no collision process.
If we notice the acceleration of in the above latter case, the relation of a a . Therefore, this indicates that the impossibility of discrimination between the particles A and B yields or between i   t t t t     , as can be seen from the expression of (5).
In the present study, we thus accept the imaginary time it   as an essential characteristic of a micro particle caused by the impossibility of discrimination between micro particles.In a collision problem, the acceleration is meaningless, although t a  is finite at the limit of and 0

Diffusion Equation of Imaginary Time
Rewriting the concentration of diffusion particles into a quantity of state expressed by a complex function   , (1) is presented as: Assuming can be solved by the separation method of variables.Using complex numbers j k , determined from the initial and boundary conditions, the general solution of ( 6) is obtained as; , , , t x x x  and using the real function 3 , we rewrite the complex function  into the complex-value function yielding Further, substituting (8) and   2 i  into (6) and multiplying the both-side of (6) by , (1) is rewritten as:

Diffusion Coefficient of Micro Particle
  , j j f t r The function is defined as a probability density which a diffusion particle in the initial state of exists in the state of  j j after j times jumps.A diffusion particle moves at random and it is, therefore, considered that the jump frequency valent in probability to their mean values of all diffusion particles in the collective system.Since it is also considered that the probability of diffusion-jump from the state t r the same state to , the relation of is thus valid.The Taylor expansion of the left-hand side of (10) yields The Taylor expansion of the right-hand side of (10) also yields The substitution of (11) and ( 12) into (10) gives Since the probability density function f of a diffusion Copyright © 2013 SciRes.JMP particle corresponds to the normalized concentration C, the comparison of (1) with (13) gives the diffusion coefficient yielding as a relation satisfying the well-known parabolic law [5].

Diffusion Coefficient and Angular Momentum
When a micro particle randomly jumps from a position to another one, the jump orientation becomes the spherical symmetry in probability.Using the equation of L r p     relevant to the angular momentum L r p   defined by a position vector r and a momentum p m v  , the right-hand side of ( 14) is re-written as: is valid in the spherical symmetry space.Considering the eigenvalue, the relation of ( 14) is thus rewritten as an operator relation of Substituting (15) into (9) gives Here, if we define the relation given by (16) becomes the equation of i 2 In mathematics, it was clarified that we can transform the (18) Further, the substitution of (3) into (18) yields the well-known Schrödinger equation ( 2).The defined equation (17) is one of the basic operators in quantum mechanics.
Hereinbefore, the Schrödinger equation was reasonably derived from the diffusion equation.It was also found that the diffusivity corresponds to the angular momentum operator in quantum mechanics.The relation of ( 15) is concretely investigated in the following section.

Discussion and Conclusion
diffusion equation for the collective motion of micro particles into the Schrödinger equation for a micro particle.In physics, energy E, momentum p and angular momentum L are expressed as operat yielding ors We cannot observe imaginary physical quantities.Theretioned in a collision problem, the im fore, the eigenvalues of their operators are meaningful in quantum mechanics.
As previously men possibility of identification between micro particles corresponds to introducing the imaginary time it   into those motions and also it corresponds to yieldin meaningless acceleration.It is considered that the physical concept obtained here is generally valid for the micro particle motions.The e ctive system of heat quantity Q and absolute temperature T is given by the well-known Boltzmann factor of where B k is the Boltzmann constan rri l depe t [6].There is an energy ba er for a diffusion particle in order to jump from a site to another site.Therefore, it is necessary for a diffusion particle to obtain the activation energy Q from the thermal fluctuation.In a collective system composed of micro particles, the diffusion coefficient D is thus directly proportional to the probability factor of (19).
The jump of a diffusion particle in a solid crysta nds on a factor  derived from the atomic configuration and on the en opy S derived from an elastic strain.In a solid crystal, therefore, (15) is rewritten as cond t cular or the stant an he mole atomic weight.Here, (20) was obtained as a new representation of diffusion coeffi-clarified.We revealed that the diffusion coefficient D in classical mechanics corresponds to the angular moment L  in quantum mechanics.The physical constant of A. Fick, Philosophical Magazine Journal of Science, Vol. 10, 1855, pp. 3