Stochastic Models of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 162

7.6.2 Schrödinger's Equation with Non-zero Potential

and Its Equivalence to Diffusion with Drift

In Sect. 7.2 the equivalence between Wiener process and the diffusion process was
demonstrated. Next, the direct relation between diffusion and quantum mechanics
will be shown [ 56 ]. The basic equation of quantum mechanics is Schrödinger's
equation, i.e.

i @

@t D H .x;t/

(7.48)

2 is the probability density function of finding the particle at position
x at time instant t, and H is the system's Hamiltonian, i.e. the sum of its kinetic and
potential energy, which is given by H D p 2 =2m C V , with p being the momentum
of the particle, m the mass and V an external potential. It holds that

where j .x;t/j

p 2
2m D

@ 2
@x 2

1

2 m

(7.49)

thus the Hamiltonian can be also written as

@ 2

1

2 m

H D

@x 2 C V:

(7.50)

The solution of Eq. ( 13.1 ) is given by [ 34 ]

.x;t/ D e iHt .x;0/ (7.51)

A simple way to transform Schrödinger's equation into a diffusion equation is
to substitute variable it with t. This transformation enables the passage from
imaginary time to real time. However in the domain of non-relativistic quantum
mechanics there is a closer connection between diffusion theory and quantum
theory. In stochastic mechanics, the real time of quantum mechanics is also the
real time of diffusion and in fact quantum mechanics is formulated as conservative
diffusion [ 56 ]. This change of variable results in the diffusion equation

1
2

V.x/

2
@ 2 @x 2

@
@t D

(7.52)

Equation ( 7.52 ) can be also written as @ @t DH, where H is the associated
Hamiltonian and the solution is of the form .x;t/ D e tH .x/, and variable
2 is a diffusion constant. The probability density function satisfies also the
Fokker-Planck partial differential equation

1

@x u .x/

@
@t D 2 2 @ 2

@

@x 2

(7.53)