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Derivation of Schrödinger’s equation
Mikhail Batanov1, Associate Professor
Department 207, Moscow Aviation Institute,
Moscow, Russia
Abstract
The base concepts of quantum mechanics (such as the matter waves of deBroglie, the Heisenberg
Uncertainty Principle, the lack of definite size and trajectory of elementary particles, the universality of
Planck’s constant and Schrödinger’s equation) still lack sufficient logical justification. The interest in
the origins of quantum mechanics is caused, among others, by the fact that the cutting edge of science in
the study of structural organization of matter, string theory, is based on quantum mechanics. Yet there
are, in my opinion, virtually insurmountable difficulties in this theory. This makes it necessary to rethink
the foundations of quantum physics.
In this article, a model of a material particle in chaotic motion (while maintaining its size and trajectory) is presented. On the basis of this model, the following is achieved:
 to express Planck’s constant ħ through the main features of a stationary random process;
 to justify the transition from the coordinate representation of the state of the particle to its momentum representation without invoking the idea of the existence of de Broglie waves or the
Heisenberg uncertainty principle.
 to derive Schrödinger equation on the basis of the principle of extremum of the mean of the action of a particle in chaotic motion.
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[email protected]
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At the same time, the conditions and limits of application of the generalized Schrödinger equation to describe phenomena on microscopic and macroscopic scales are highlighted.
An intermediate result, the determination of the density of the probability distribution of the n-th
derivative of an n-th order differentiable, random, stationary process, can be applied in many areas of
probability theory and statistical physics.
Keywords: Schrödinger equation, Planck's constant, electron core, probability density distribution, derivative of a random process, particle, chaotic trajectory, coordinate representation.
1. A brief history of the emergence of Schrödinger’s equation
One of the main mysteries of quantum mechanics, and hence of all modern extensions of this
theory, is that of the emergence of Schrödinger’s equation. The lack of logical justification for the derivation of the original equation negatively affects the development of our understanding of the structural
organization of matter.
Schrödinger’s equation takes the following form:
i

2 2

   U ( x, y, z ) ,
t
2m
(1)
where  =  (x, y, z, t) is the wave function describing the state of an elementary particle, U (x, y, z) is
the potential energy of the particle,  is the reduced Planck's constant, and m is the mass of the particle.
It is believed that this equation was obtained by Erwin Schrödinger (1887 – 1961) on the basis of inductive and deductive assumptions established by 1926 as a result of experimental studies of the properties
of elementary particles.
Of particular importance at that time was the idea, conceived by Louis de Broglie (1892 - 1987),
that material particles in motion could possess wave properties. In his doctoral thesis, Recherches sur la
théorie des quanta (Research on the Theory of the Quanta, 1924), Louis de Broglie compared the rectilinear trajectory of the free motion of a particle with a direct ray of light, and came to the conclusion that
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they are described by the same Jacobi equation, arising from the fundamental principle of "extremum of
action". It turned out that the trajectory of the free motion of the particle and the beam of light are extrema for virtually the same functional, the action. This circumstance prompted Louis de Broglie to suggest that if the wave described by the equation
w = expi(t – kr),
(2)
(where  is the angular frequency; k is the propagation vector; t is time; r is the dimensional vector),
displays some properties of a particle, it is possible to reverse the assertion to say that a material particle
in motion can correspond to a plane wave described by
 = expi(Et – pr)/,
(3)
where E is the kinetic energy of a moving particle, and p = mv is its momentum.
In addition, Louis de Broglie was acquainted with experiments, carried out by his elder brother
Maurice de Broglie, which were associated with the physics of X-ray radiation, as well as with the pioneering work of Max Planck and Albert Einstein on the quantum nature of radiation and absorption of
light. This allowed him in 1923 to 1924 to propose that a moving particle can be associated with an oscillatory perturbation  having frequency
 = E/
(4)
 = 2/|p|.
(5)
and with the wavelength
This idea was supported by P. Langevin and Einstein, but most of the physics community reacted to it
with skepticism. However, in the period from 1927 to 1930, several groups of experimenters (C. Davisson & L. Germer, and O. Stern, I. Estermann et al.) showed that the idea of the existence of matter
waves, proposed by de Broglie, could be used to describe the phenomenon of the diffraction of electrons
and atoms in crystals.
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In one of his early works of 1925 to 1926, Erwin Schrödinger, critical of the Bose-Einstein statistics formulation, wondered, "Why not start with the wave representation of the gas particles, and then
impose on such ‘waves’ the quantization conditions ‘à la the Debye model’"? After that followed his
central idea: "This implies none other than the need to take seriously into consideration the proposal of
L. de Broglie and A. Einstein concerning the wave theory of moving particles."
The next article by Schrödinger already contained equation (1), marking the beginning of the intense development of quantum mechanics, together with the great work of Max Planck, Albert Einstein,
Niels Bohr and Werner Heisenberg.
The arguments contained in the derivation of Schrödinger’s equation (1) were subsequently recognized by experts to be invalid, but the equation itself was deemed correct. This was not the only such
case in science. For example, the fundamental equations of electrodynamics were derived by James
Clerk Maxwell from incorrect assumptions about the mechanical properties of the ether.
In the last ninety years since 1926, many researchers have offered various ways to justify Schrödinger’s equation. But as far as I know, none of these attempts succeeded. The basics of quantum mechanics to this day provoke debate in the scientific community.
There have been so many unsuccessful attempts to derive the Schrödinger equation that it has
become obvious that to achieve this result in the framework of quantum - mechanical logic is impossible.
In this present paper, a model of a particle (having a given volume and continuous trajectory) in
chaotic motion clearly contradicts the corresponding neo-positivist views, but it leads to the derivation
of the Schrödinger equation, as you are invited to convince yourself from the rest of this paper.
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2. Model of particle in motion
Consider a particle occupying a small volume
compared to that of its surrounding space.
(Figure 1) By convention, we call this particle a
"point". Suppose that this "point" is in constant chaotic motion around the reference point (serving as the
origin of the coordinate system XYZ) under the influence of various mutually independent force factors.
Examples of "points" in continuous chaotic motion
may be atoms vibrating in a crystal lattice, flies flying
Fig. 1. The particle (“point”) in chaotic
motion in the neighborhood of the central
reference point so that its total mechanical
energy E remains constant.
in a jar, the nucleus vibrating inside a biological cell,
a human embryo moving in the womb, tips of branches fluttering in the wind, and so forth.
In what follows, we suppose that such a chaotic motion of the "point" continues "forever" because its
total mechanical energy E always remains constant:
Е = T (x,y,z,t) + U (x,y,z,t) = const,
(6)
where T (x, y, z, t) represents the kinetic energy of the "point" due to its velocity, and U (x, y, z, t) is the
potential energy of the "point" associated with a force tending to return the "point" to the central reference point (i.e., with an elastic force).
Thus, in this model, each of the energies T (x, y, z, t) and U (x, y, z, t) of the "point" is a random
function of time and its position relative to the "center". But these energies flow smoothly into each other so that their sum (i.e., the total mechanical energy E) always remains constant.
If the speed of the "point" in chaotic motion in the vicinity of the central reference point (Figure
1) is low, then according to non-relativistic mechanics, it has kinetic energy
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T ( x, y, z, t ) 
p х2 ( x, y, z, t )  p 2у ( x, y, z, t )  p z2 ( x, y, z, t )
2m
.
(7)
For brevity, instead of (7) we write
T (t ) 
p х2 (t )  p 2у (t )  p z2 (t )
2m
,
(8)
where pх(t), pу(t), pz(t) are the respective instantaneous values of the spatial components of the momentum of the "point" in chaotic motion,

p
p х2  p 2у  p z2 ,
pi  mvi  m
dxi
 m  xi .
dt
(9)
(10)
The type of potential energy U (x, y, z, t) acting on the “point” is not specified.
The action of the "point” S under consideration is defined in non-relativistic mechanics as follows (Landau & Lifshitz 1988)
t2
S (t )   [T ( p x , t )  U ( x, t )]dt  Et.
(11)
t1
To simplify the calculations, we consider the one-dimensional case, without loss of generality. The
three-dimensional case merely requires more integrations.
Due to the complexity of the path of the “point" in motion, we are interested not in the action itself (11), but rather its average over time (resp., over its realizations).
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S  lim
N  N
t2
N
 S (t )  [T ( p , t ) U ( x, t )dt  E t.
i
i 1
x
t1
(12)
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Recall that for an ergodic stochastic process, an average
over time is equivalent to the average over its realizations.
The plus sign in the integrand of (12) results from
the fact that the mean potential energy is negative, as the
“point” always tends to return to the reference point at the
center of the sphere formed by the path of the point over
time. (Figure 2). Finding the mean (12) is carried out over
the realizations, taken for the same period of time
Fig. 2. The total path of a particle in
continuous chaotic motion as it tends
towards spherical symmetry.
 t = t2 – t1 .
The mean kinetic energy of a "point" in chaotic
motion may be represented as
1
T ( px , t ) 
2m

  ( p ) р dр
x
2
x
x
,
(13)

where ρ(px) is the probability density function of the momentum component px of the material "points".
The mean potential energy of a "point" may be represented as

U ( x, t )    ( x)U ( x)dx ,

(14)
where ρ(х) represents the probability density function of the projection onto the x-axis of the “point” in
chaotic motion around the fixed reference point. (Figures 1 and 2).
Substituting (13) and (14) into the equation (12) for the mean of the action, we obtain
t2 



 1
2
S  
 ( p x ) р x dрx   ( x)U ( x)dxdt  E t.
2m

t1 






(15)
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For the further derivation of the Schrödinger equation, two auxiliary points are explained below.
The first point, developed by the author (Gaukhman 2007 / 2008)), is dedicated to the definition of the
distribution of the probability density of a stationary stochastic process which is n-times differentiable.
The second item, the "coordinate representation of the average momentum of a particle" is borrowed
from the work of D. Blokhintzev (1963) for ease of reference.
3. Determination of the probability density function of the n-th derivative
of an n-times differentiable stationary stochastic process
The key to the understanding of quantum mechanics and the limits of its application lies in the determination of the probability density function (PDF) of the derivative of a strictly stationary stochastic
process, given that the PDF of the process itself is already known. This solution makes it possible to justify the quantum-mechanical transformations between a coordinate and a momentum representation
without appealing to the concept of de Broglie waves. This is made possible due to the fact that the momentum of a material “point” is linearly related to the derivative of its coordinate px= m·x/t = mx‫׳‬.
Moreover, the problem of determining the one-dimensional PDF ρ1[ξn(t)] , that is the n-th derivative of the n-times differentiable stationary stochastic process ξ (t), given only that its one-dimensional
PDF ρ1[ξ (t)] is known, arises in a series of problems in the fields of radio physics and statistical mechanics.
We first consider the general properties of the first derivative of the stationary stochastic process
ξ (t) by regarding its realizations (Figure 3).
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Fig. 3. Representation of the differentiable stationary stochastic process ξ(t).
Figure 3 shows that the value of the random variable ξ (ti) at the interval ti is independent from the
derivative  (t i ) 
 t1 
of this process taken at the same value ti, ; that is, they are random variables
t1
which are not correlated. This may be expressed analytically (Ritov 1976):
  ti  ti  
d 1
 ti 2  1 d   ti 2   0,
dt 2
2 dt
(16)
where   denotes taking the mean over the realizations. This result follows from the facts that the differentiation and averaging operators commute in this specific case, and that the average characteristics
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of the strictly stationary process, including its dispersion, are constant over time:   t i    const
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Realizations of a stationary stochastic process ξ(ti), such as that shown in Figure 3, may be interpreted as the change over time of the projection onto the X-axis of the position of the “point” in motion
(Figures 2 and 3). That is, x(t) = ξ(ti).
Even in the case of the statistical independence of the random values ξi and ξI, there exists a connection between the PDF’s ρ1(ξi) and ρ1(ξi ). This results from the well-known procedure of obtaining
the PDF of a derivative ρ1(ξi) for a given two-dimensional PDF of a stationary stochastic process (Ritov
1976, Tihanov 1982).
 2  i ,  j    2  i , t i ;  i , t j  .
(17)
This is done by changing variables in (17):



 i   k   k ;  j   k   k ; t i  t k  ;
2
2
2
t j  tk 

2
,
(18)
where
  t j  ti ; t k 
t j  ti
2
with the Jacobian transform [J] = τ. Then, from the PDF (17) we obtain





 2  k ,  k   lim   2   k   , t k  ;  k   , t k   .
 0

2
2
2
2
(19)
Further, integrating the resulting expression over ξk , we find the desired PDF of the derivative of the
original process over the interval tk (Ritov 1976, Tihanov 1982):

1 ( k )    2 ( k ,  k )d k .

(20)
The formal procedure given by (17) through (20) solves the problem of determining the PDF ρ1(ξ )
for a given two-dimensional PDF (17). However, a two-dimensional PDF is defined only for a very lim-
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ited class of stochastic processes. It is therefore necessary to consider the possibility of obtaining a PDF
ρ1(ξ ) for a given one-dimensional PDF ρ1(ξi).
To solve this problem, we use the following properties of stochastic processes:
1] A two-dimensional PDF of any stochastic process can be represented as (Ritov 1976, Tihanov 1982)
 2  i , t i ;  j , t j   1  i , t i   j , t j /  i , ti  ,
(21)
where ρ(ξj, tj /ξi, ti) corresponds to the given PDF.
2] For the strictly stationary stochastic process, the following identity holds (Ritov 1976, Tihanov 1982)
1  i , ti   1  j , t j 
(22)
3] The given PDF of a stationary stochastic process as ti tends to tj degenerates into a delta function (Tihanov 1982)




lim  2  i , t i /  j , t j    i   j .
 0
(23)
Based on the above properties, we consider a stochastic process over the interval ] ti – τ0; ti + τ [
as τ tends to zero, using the following formal procedure. The PDF’s ρ1(ξi) = ρ1(ξi, ti) and ρ1(ξj) = ρ1(ξj, tj)
can always be represented as the product of two functions:
 ( i )   i  i    2  i 
(24)
 ( j )    j   j    2  j  ,
where φ(ξi) represents the wave function of a random variable ξi over the interval ti.
For a stationary stochastic process, we have the identity
  i     j  ,
(25)
as is easily seen by taking the square root of both sides of (22). Then, according to (24), we obtain (25).
Note that identity (25) is approximately true for the majority of non-stationary stochastic processes as τ
tends to zero, that is,
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  i , ti   lim   j , t j  ti   .
(26)
 0
When the condition (25) holds, equation (21) can be represented in the symmetric form
  i ,  j     i   j /  i    j  ,
(27)
where ρ(ξj /ξi) is the given PDF; in expanded form this becomes



  i , t i  t k  ;  j , t j  t k   
2
2


 

 


   i , t j  t k     j , t j  t k 
/  i , t i  t k    i , t i  t k   .
2 
2
2 
2

(28)
Setting τ to zero in (28) in such a way that the interval τ contracts in equal measure on the left and the
right to a point tk = (tj – ti)/2, then taking into account (23) and (27), we obtain


lim  ( i ,  j )  lim  ( i )  ( j /  i ) ( j )   ( ik ) ( ik   ik ) ( jk ),
 0
 0
(29)
where ξik is the result of the stochastic value ξ(ti) tending to the stochastic value ξ(tk) on the left, while ξjk
is the result of the stochastic value ξ(tj) tending to the stochastic value ξ(tk) on the right.
Integrating both sides of the expression (29) over ξik and ξjk,, we obtain


   ( ik ) ( jk   ik ) ( jk )d ik d jk  1 .
 
(30)
Expression (30) is a formal mathematical identity out of the theory of generalized functions, taking into account the properties of the delta (δ)-function. In order to assign the expression (30) a physical
content, one must specify the specific type of δ-function involved.
Therefore we now fix the form of a δ-function for a Markov stochastic process. Consider a continuous stochastic Markov process which satisfies the Einstein-Fokker equation (Tihanov 1982, Wentzel
& Ovcharov 1991)
 ( j /  i )
t
B
 2  ( j /  i )
 2
,
(31)
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where B is the diffusion coefficient. This parabolic differential equation has three solutions, one of
which can be represented as (Tihanov 1982, Wentzel & Ovcharov 1991):


1
 2  j , t j /  i , ti 
2

 exp{iq(
j
  i )  q 2 B(t j  ti )}dq,
(32)

where q is the generalized parameter. As tj – ti = τ tends to zero, then from (32) we obtain one of the definitions of a δ-function
1
lim  ( i /  j ) 
 0
2

 exp{iq(
jk
  ik )}dq   ( j   i ),
(33)

Since this result was obtained for the limiting case, that is, as τ tends to zero, then it is possible
that the δ-function (33) can conform not only to Markov, but also to many other stationary stochastic
processes.
Substituting the δ-function thus obtained (33) into (30) yields


 
 
1
 ( ik )  ( jk )
2

 exp{iq(
jk
  ik )}dq d ik d jk .
(34)

Changing the order of integration in (34), we obtain
 1

 2
 




 ( ik ) exp{ iq ik }d ik

1
2



 ( jk ) exp{iq jk )d jk dq .



(35)
Take into account that, according to (25), φ(ξik) = φ(ξjk), and also the properties of δ-functions,
ξik = ξjk = ξk. Under these conditions, the expression (35) takes the form

  (q)

where

( q ) dq  1,
(36)
14
 q  
1
2
1
 q  
2
*

 (
k ) exp{ iq k }d k ,
(37)


  (
k
) exp{iq k }d k .
(38)


The integrand  (q) (q) in (36) satisfies all the requirements of the PDF ρ(q) of the random vari-
able q:
 (q)   (q)  (q)   (q) .
2
(39)
We now investigate the random variable q. We first reconsider (32). The result of the integration
on the right side of this expression does not depend on the variable q. Therefore it may be considered as
a sort of generalized frequency. However, both the physical statement of the problem as well as the
mathematical formalism in the expression (32) impose on q the following restrictions:
1) The variable q must characterize a stochastic process in the interval under investigation
] ti – τ; ti + τ [ as τ tends to zero;
2) The variable q, according to the mathematical notation on the right side of (32), must belong to the
set of real numbers (q R), having the cardinality of the continuum; that is, it must have the possibility to take any value in the range ]– ∞, ∞ [;
3) q must be stochastic
All these requirements are satisfied by any of the following random variables associated with a
stochastic process in the interval under investigation τ:
i 
 k
 2 k
 n i
,  i 
, . . . ,  n i 
.
t
t
t n
(40)
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But these random variables do not equally characterize the process. We consider one of the realizations of the test process. The function ξ(t) (Figure 3) in the interval ]ti ; tj = ti + τ[ for τ < τcor (where
τcor is the correlation radius of a stochastic process) may be expanded as a Maclaurin series:
 t j    ti    (ti ) 
 (ti )
2
 2  ... 
 (n)
n!
 n  ...
(41)
 ...
(42)
We rewrite the expression (41) in the form
 j  i

 i 
i
2!
  ... 
 ( n) n1
n!
As in (33), we let  tend to zero, whereby (42) reduces to the identity
lim
 j  i
 0

  k
(43)
In this way, the only random variable satisfying all the above-mentioned requirements in the interval under investigation ]ti = tk – τ/2; tj = tk + τ/2[ as τ tends to zero is the first derivative of the original
stochastic process ξ'k over the interval tk. Therefore we may assume that the random variable q in (32)
through (39) is directly proportional to ξ'k, that is
q
 k
,

(44)
where 1/η is the proportionality coefficient.
Substituting (44) into (36) through (39), we obtain the following procedure as required for determining the PDF of the derivative ρ(ξk) of a stationary Markov process ξ(t) over the interval tk , given the
one-dimensional PDF ρ(ξk) over the same interval. (This procedure may be valid for other stochastic
processes as well.)
1] Express the given one-dimensional probability distribution ρ(ξ) as the product of two wave functions
φ(ξ):
16
(  )       .
(45)
2] Two Fourier transforms are then carried out:

1
 ( ) 
2
  ( ) 
  ( ) exp{i  / }d ,
(46)


1
2
  ( ) exp{ i  / }d .
(47)

3] Finally, for any given interval of a stationary Markov process we get the desired derivative of the
PDF, as follows:
 ( )     *       .
2
(48)
As we have already remarked, the procedure given by (45) through (48) can be applied not only to
the stationary Markov processes, but to many other stationary processed for which the δ-function in (30)
takes the form (33) as τ tends to zero.
To find the physical interpretation of the proportionality factor 1/η, we make a comparison to
known results. This method is not mathematically perfect, but permits us to quite efficiently obtain a
sound result of practical importance.
We consider a stationary Gaussian stochastic process. Over every interval of the process, the random variable ξ obeys the Gaussian distribution:
 ( ) 
1
2 
2


exp  (  a ) 2 / 2 2 ,
(49)
where ξ2 and аξ are the variance and expected value, respectively, of the process ξ(t).
Subjecting the PDF (49) to the sequence of operations (45) through (48), we obtain its derivative:
17
 ( ) 
 ( ) 

1
2  / 2 

1
2  / 2 
2

 2

exp 

 2  / 2 


2 ,



(50)
2

 2

exp 

 2  / 2 


2 ,


(50)



On the other hand, with the help of a well-known procedure indicated in (17) through (20) for an
analogous case, we obtain (Ritov 1976, Tihanov 1982)
 ( ) 
1
2
2



exp    2 / 2 2 ,
(51)
where σξ’ = σξ /τcor, and τcor is the correlation radius of the initial stochastic process ξ(t).
Comparing the expressions (50) and (51) we find that

2 2
 cor
(52)
These PDF’s are in complete agreement with one another.
Expression (52) is obtained for a Gaussian stochastic process, with σξ being the standard deviation and τcor being the correlation radius, the basic characteristics of a stationary stochastic process.
The rest of the initial and central moments in the case of a non-Gaussian distribution with the random
variable ξ (t) will give a small but insignificant contribution to the expression (52), so it can be asserted
with a high degree of confidence that this expression is applicable to a wide class of stationary stochastic processes.
It should be noted that in statistical physics and quantum mechanics, the transition from the coordinate representation of a function of an elementary particle state to its momentum representation is
effected by a formal process almost completely analogous to the procedure (45) through (48). The difference is only in determining the proportionality factor 1/η.
18
In quantum mechanics it is well known that if the projection onto the x-axis of the position of a
free elementary particle (e.g., an electron) is described by a Gaussian distribution (Landau & Lifshitz
1989)
 ( х)   х  
1
2
2 х2
2

 x 

exp 
2,

 2 x 

(53)
where σx is the standard deviation of the projections of the positions of an elementary particle onto the
x-axis in the neighborhood of the mean (that is, the “center” of the system). Then, as the result of operations analogous to that of (45) through (48), it turns out that the PDF of the momentum component рх of
an elementary particle is also Gaussian (Landau & Lifshitz 1989)
 ( px )    px  
1
2
2 2px

p x2

exp 
2

 2 px





(54)
with the standard deviation
 px 

2 x
,
(55)
where the reduced Planck constant  = 1.055·10-34 J/Hz.
If we now take into account that the momentum component of an elementary particle (e.g. an
electron) px is equal to
p x  me
dx
 me x ,
dt
(56)
where me is the electron rest mass, then taking into account (55), the PDF (54) becomes
 ( x) 
1
2  /(me 2 x )2


x 2


exp 
2.

 2 /(me 2 x ) 

(57)
19
Comparing (50) to (57), while taking into consideration (52) and that ξ' = х', we find that for the
given case

2
2 ex
 ex


,
me
(58)
where
 ex
2
2me ex
2  0,91 10 30
2
2


  ex
 1,73 10 4  ex
 34

1,055 10
(59)
is the correlation radius of a stationary stochastic process, which is the result of the projection of the stochastic motion of the "point" (e.g., an electron) onto the x-axis near the stationary "center" of the system
(Figures 1 and 2).
σex is the standard deviation of the projection onto the x-axis of the “point” (e.g., an electron) in
chaotic motion around the mean (that is, the relative “center” of the system).
From (58) it follows that the Planck constant is not a fundamental physical constant, but rather a
random variable which may be expressed in terms of more basic parameters of a stationary stochastic
process.

2 2par, x m
 par, x
,
(60)
where for a general case:
σpar,x is the standard deviation of the projection onto the x-axis of a particle ("point") in chaotic
motion in the neighborhood of the point of reference (i.e., the “center” of the system).
τpar,x is the correlation radius of a given stationary stochastic process.
For a wide range of applications, the expression (60) is in itself very important, as is the related ratio (52), which in the general case may conveniently be represented as follows:
20
ч 
2 2par, x
 par, x


m
(61)
Note the following interim conclusions:
1] The quantum-mechanical transition from the coordinate representation to the momentum one is not
only applicable to the processes in the world of elementary particles, but also to any stationary Markov
stochastic processes (and probably many other stochastic processes), both in the microcosm and in the
macrocosm. For example, a tree branch, constantly moving chaotically around its middle position (the
point of reference serving as the "center") by the rapidly changing direction of wind gusts, behaves similarly to elementary particles in the "potential well". The fluctuations of these movements of the branch
would also have a discrete (quantum) average set of states, depending on the intensity of the wind gusts.
With weak wind gusts, the branch generally fluctuates near the central reference point, in a way that the
position of its tip can be described by a Gaussian distribution. Wind gusts of greater intensity will see
the tip of the branch rotate in a circular motion; at still higher intensities, it would essentially describe a
figure eight, and so forth. Depending on the strength of the wind, the tip of the branch may describe a
discrete set of Lissajous figures. In other words, the quantum-mechanical formalism is not an exclusive
feature of the microcosm; it is also applicable to the statistical description of many stochastic processes
in the macrocosm.
2] The algorithm described by (45) through (48) of the transition from the coordinate representation
ρ(ξi) to the momentum representation ρ(mξi) and vice versa was obtained in the context of a particular
δ-function (33). It would be interesting to analyze what would be the result in the case of other types of
δ-function.
3] On the basis of the foregoing, we may obtain the PDF ρ(ξi''), which is the second derivative of a differentiable stochastic process which is differentiable to at least order two. In this case, we should con-
21
sider not the process  (t) itself, but its first derivative (t) = (t)/t. Then the distribution of the second derivative may be obtained according to the same procedure, (45) through (48), whereby
ρ(ξi).would replace ρ(ξi) in those equations.
Analogously, we also may obtain the PDF ρ(ξi(n)) of any derivative of n-times differentiable stationary stochastic process with the help of the following recursive procedure:



 ( (n1) )    n1   n1 ;
 
  n  
2
 
  n  
*

1
  (
( n1)

1
2

 i ( n ) ( n1) 
 ( n1)
) exp 
d
;
n






(62)
 i ( n) ( n1)  ( n1)
 d
;
n


 ( ( n1) ) exp 

(63)
(64)
   
 ( (n) )    n  *  n ,
where
n 
2 2( n 1)

 cor ( n 1)
(65)
where   ( n 1) ,  cor  n 1 are the variance and the correlation radius, respectively, of the given n–1 times
2
differentiable stationary stochastic process.
4] The procedure (45) through (48), fully analogous to that of the quantum-mechanical transition from
the coordinate representation of a quantum system to its momentum representation, is obtained here on
the basis of the study of realizations of a common stationary stochastic process, that is, without the use
of phenomenological principles of wave-particle duality.
5] There is no need to use the de Broglie conjecture of the existence of matter waves to describe the diffraction of atoms and electrons on the crystal lattice. One may study paragraph 2.9.6 of (Gaukhman
22
2008), where one will find the derivation of the formula to create a three-dimensional diagram (indicatrix) of the scattering of particles on the surface of a multilayer periodic crystal:
 ( ,  / ,  )  4
n12 k к
sin 2 [ n1 / 2  k к (a 2  b 2 ) / c 2 / 2] ca b  a b   c ba  ab 

,
[( n1 ) 2  k к2 (a 2  b 2 ) / c 2 ]2
c2 a2  b2
(66)
where
a  cos cos  cos cos , b  cos sin   cos sin  , c  sin  sin  , av   sin cos ,
bv   sin v sin , cv  cos v , a   cos v sin  , b  cos v cos
Angles ,  , ω and ν are shown in Figure 4;
kк = rcor n11/2/ (0,066 l1 ),
where:
l1 signifies the thickness of one layer, i.e. one sinusoidal equipotential surface;
n1 equals the number of layers effectively involved in the scattering of the particles;
rcor is the average radius of curvature of a sinusoidal equipotential surface.
For a single crystal, all the sinusoidal equipotential surfaces have the same, that is, rcor. Therefore in this
case rcor signifies the effective cross section of the scattering of electrons by the atoms of a crystal.
The results of a calculation using (66) at an angle of incidence of particles on the surface of the
crystal using the values  = 45о, azimuth angle  = 0о, rcor = 0.0000000001 = 10–10 cm, l1 = 0.000000001
= 10–9 cm, n1 = 1940 (layers) are shown in Figure 4:
23
Fig. 4. Diagram (indicatrix) of particle (electron) scattering through 1940 layers
of sinusoidal equipotential surfaces of a crystal, calculated according to
(66) using MathCad14 (Gaukhman 2008, Batanov 1994).
4.The coordinate representation of the average particle momentum
The contents of this paragraph are well known to specialists in the field of quantum mechanics.
However, given the importance of subsequent conclusions and for ease of reference, the following calculations are almost completely rewritten from (Blokhintsev 1963).
Let us first recall the properties of the Dirichlet integral appearing in the theory of Fourier integrals
and the theory of generalized functions (Blokhintsev 1963):
0, if a, b  0 or a, b  0,
1b
sin kz
dz  
  ( z)
z
k   a
 0, if a  0, b  0 .
(67)
1 sin kz
  z 
k  
z
(68)
lim
whereby
lim
This is one of many forms of a δ-function.
We now consider the calculations for reducing dimensions by demonstrating the equality
(Blokhintsev 1963):
24



 
pxn    ( px ) pzn dpx    ( px ) pzn ( px )dpx    ( x, t ) i   ( x, t )dx,
 x 



n
(69)
where the exponent n is a positive integer. Taking the mean over time (or realizations) to the power of n
momentum components
рхn = (m·x/t)n = (mx‫)׳‬n;
(70)
where  x  and  ( p ) are the wave functions which were introduced in (24).
х
[ x    x  ] and (48) [  ( px )    px    mx ], and, following (46) and (47), are related to one
other (assuming a stationary stochastic process) via a Fourier transform:
i

  p x  mx   ( х)

xx
i

ч
px x

е
е
dx   ( х)
dx ;
1/ 2
1/ 2
(2 )
(
2

)

i

xx
ч


i
i
(71)
px x
е
е 
 *  p x  mx   ( х)
dx


(
х
)
dx ;

(2 )1/ 2
(2 )1/ 2



 x    ( x)

i
xx
ч
px x

е
е
dx   ( р х )
dр х ;
1/ 2
1/ 2
(2 )
(
2

)

i

xx

ч
i
(72)
px x
е
е 

 * x    ( x)
d
x


(
р
)
 х (2 )1/ 2 dрх ,
(2 )1/ 2


where the parameter ηpar is defined by (61)
 par 
2 2par, x
 par, x


.
m
(73)
In order to prove (69), for ψ(рx) and ψ*(рx) we substitute their respective expressions in terms of
integrals (71) (Blokhintsev 1963):
25
p x
p x
i x k
i x l





e
e
n
n

px    ( xk )
dxk  px  ( xl )
dxl  dpx .
1/ 2
1/ 2


(2 )
(2 )
 



 
(74)
An immediate consequence of this is that
n
x
p e
i
p x xl

 
  i
 xl
n
 i px xl
 e
.

(75)
Substituting (75) into (74) we obtain:
1
p 
2
n
x
p x


i x k
 





(
x
)
e
dx
k   ( xl ) i
  k
 xl


 
n
 i px xl 
 e
dxl dp x .


(76)
We perform an integration by parts n times, starting from the second integral in the integrand. In doing
so, we assume that ψ (x) and its derivatives vanish at the boundaries of integration, that is, at x = ± ∞.
Following these steps, we find (Blokhintsev 1963):
1
p 
2
n
x
p x
p x


i x k
i x l 


  ( xk )e dxk  e   i xl

 
n


  ( xl )dxl dp x


(77)
Changing the order of integration, we first integrate over px (Blokhintsev 1963):


 
1
p 
  ( xk )dxk   i

2 
xl
 
n
x
n
p ( x x )

i x k l

  ( x j )dxl  e  dp x .


(78)
We introduce the variables ξ = рx /h, z = xk – xl. In the last integral in (78), we perform the integration over ξ between the limits – k to + k; then passing to the limit
k → ∞, this expression takes the form (Blokhintsev 1963):


   n

sin kz
p xn    i   ( x) dx  lim   ( x  z )dz

k 

x

z







n

  
 
   i   ( x)dx   ( x  z ) ( z )dz.
 

 x 
(79)
26
Based on the properties of the Dirichlet integral (67), when a = –∞; b = + ∞, and
ψ(z) = ψ*(x + z), we have (Blokhintsev 1963):
n


   n

  
p xn    i   ( x)  ( x)dx    ( x) i   ( x)dx
 x 



 x 
(80)
or

 
p   ( x, t ) i   ( x, t )dx ,
 x 

n

n
x
(81)
where
 ( x, t )   ( x) expiu
(82)
  ( x, t )   ( x) exp iu
(u is an arbitrary real number); thereby the expression (69) is proven (Blokhintsev 1963).
Using (70) and (73) from the expression (81) we obtain
n

n
 
 dx 

x n        ( x ) i par   ( x )dx .
x 
 dt 


(83)
The generalization to three dimensions then increases the number of integrations (Blokhintsev 1963).
5. Derivation of Schrödinger’s equation
We return to the average action of the particle ("point") in chaotic motion (15)
t2

 1 

2
S  

(
p
)
р
dр


(
x
)
U
(
x
)
dx
dt  E t ,
x
x
x


2
m
t1 



(84)
Imagine the action (84) in coordinate form. To do this, perform the following steps.
1] We write the PDF ρ(х) as a product of two wave functions using ψ(х,t):
 ( x)   x, t   * x, t  ,
(85)
27
 Et 
,
 
 ( x, t )   ( x) exp i
(86)
 Et 
  ( x, t )   ( x) exp  i .
 
(87)
2] We use the coordinate representation of the mean momentum, raised to the power n (81). Thus in particular we have


 
p    ( p x ) p dp x   ( x, t ) i   ( x, t )dx ,
 x 


2
x
2
z
2

(88)
3] Using (88), we represent the mean kinetic energy of the "point" (13) in the form


1 2
1
2
 2  ( х, t )
2
T
рx 
 ( p x ) р x dр x  
 ( x, t )
dx,
2m
2m 
2m 
x 2
(89)
4] By taking (85) into consideration, we can now represent the mean potential energy of the "points"
(14) in the form

U   ( x, t )U ( x)  ( x, t )dx,
(90)

5] It is then easily seen that

EЕ
 i ( x)e

iEt / 
 ( x)e iEt / 
dx  const ,
t
(91)
or, taking into consideration also (86) and (87)

E  Е    ( x, t )

  ( x, t )
dx
t
(92)
6] Substituting expression (89), (90) and (92) into (84), we may express the mean of the positions of the
particles ("point") in chaotic motion in coordinate form
28
t2


 2 
 2  ( x, t )
  ( x, t ) 
S   

(
x
,
t
)
dx


(
x
,
t
)
U
(
x
)


(
x
,
t
)
dx

i


(
x
,
t
)
dx dt
2



2
m

x

t
t1 




(93)
or
t2 
S
 2
 2  ( x , t )
  ( x, t ) 


(
x
,
t
)
 ( x, t )U ( x)  ( x, t )   ( x, t )i
2
 2m
dxdt
t

x


t1 

(94)
Finding an extremum for the mean of the action (94) requires setting the first variation to zero
t2 
S  
 2
 2  ( x, t )
  ( x, t ) 


(
x
,
t
)


(
x
,
t
)
U
(
x
)


(
x
,
t
)


(
x
,
t
)
i

2
 2m
dxdt  0
t

x


t1 

(95)
The extremum of the functional (95), that is, the function  (x, t), for which the mean of the action (95) takes an extreme value, is determined by the Euler-Poisson equation (Èl'sgol'ts 1969). For the
Lagrangian L, which is the integrand in the action functional

z z  2 z  2 z  2 z 
dxdt , where z = (x, t),
S   L x, t , z, , , 2 , 2 ,
x t t x tx 

(96)
the equation takes on the form
Lz 


2
2
2
Ls   0 ,
Lp 
Lg  2 L r   2 L t  
x
t
xt
x
t
 
 
(97)
where
p
z
z
2 z
2 z
; g
; r 2; s
,
x
t
tx
x

z
p
g
L p  L px  L pz
 L pp
 Lgp
x
x
x
x
 
(98)
29
complete the partial derivative with respect to x.
Analogously,

L
p
g
Lg  Lgt  Lgz
 Lgp
 Lgg
t
t
t
t
 
(99)
and so forth.
Using the integrand of the mean of the action (95)
L
2
 2  ( x, t )
  ( x, t )
 ( x, t )
 ( x, t )U ( x)  ( x, t )  ( x, t )i
2
2m
t
x
we define
Lz  
 2  2  x 
  x 
 2  x U  x   i
;
2
2m x
t

L p   0;
x

Lg   2i  x  ;
x
t
2
Lt   0;
t 2
2
Ls   0;
xt
2
h 2  2  x 


L


.
r
x 2
2m x 2
Substituting these expressions into (97), we obtain the desired equation for determining extrema ψ (х,t)
of the mean of the action functional (95)
i
 ( x, t )
 2  2 ( x, t )
 
+ U ( x)  ( x, t ) ,
t
2m x 2
(100)
where
 (x,t) (x,t) = | (x)| 2 = (x)
is the PDF of the projections onto the x-axis of the position of the given particles ("points") in chaotic
motion around the point of reference serving as "center" of the system in such a way that its total mechanical energy E is always constant (E = const), while the projection x(t) itself is a stationary stochastic
process. The generalization to three dimensions only increases the number of integrations; thus we have
i


 (r , t )
 2  2 (r , t )

+ U (r , t )  (r , t ) ,
2
t
2m r
(101)
30
where



 (r , t )   (r ) exp  i
Et 
,

where r is the radius vector starting at the above mentioned "center" of the configuration
(r2 = x2 + y2 + z2) (Figure 1).
Equation (101) is none other than the Schrödinger equation (1) with the Born understanding of
the meaning of the wave function. But in this case, "the Planck constant” ħ is not a fundamental constant, but rather the parameter which can be expressed in terms of the ratio of the average characteristics
of the stationary stochastic process being investigated, as above in (60)

2 2par, x m
 par, x
,
Dividing both sides of the equation (101) by ħ we get


 (r , t )
  2 (r , t )
i

+ U (r , t )  (r , t ).
2
t
2m r
Taking (61) into consideration, this equation takes the form
i
 par  2 ( r , t )


 ( r , t )

+ U (r , t ) (r , t ) ,
t
2
r 2
(102)
where
 par 
2 2par,r
 par,r
,
(103)
whereby
 par,r 
1
 2par, x   2par, y   2par, z
3
(104)
is the mean standard deviation of the positions of particles (material "points") in chaotic motion around
the point of reference acting as "center" (Figure 1), while
31
 par, r 
1
( par, x   par, y   par, z )
3
(105)
is the mean correlation (or rather autocorrelation) radius of this stochastic process.
Equation (102) is called the generalized Schrödinger equation, as it is suitable to describe the
most probable average states of point objects in chaotic motion, on microcosmic as well as on the macrocosmic scales, for a given stationary stochastic process with its total mechanical energy held constant.
Equation (102) is an equally good description of discrete sets of the average behavior of an electron in a potential well, the nucleus in the cytoplasm of biological cells, the center of the embryonic
mass in the womb, the core in the bowels of the planet, flies trapped in a jar, and so on. Each of these
stable stochastic processes has the ability to move from one steady state to another with the absorption
or release of a particular portion of its total mechanical energy.
In this way, together with the derivation of the generalized Schrödinger equation (102) we come
to the realization that quantum transitions are not unique to the atomic scale, but also are manifested in a
fractal manner at all levels of organization of matter.
The approach proposed in this paper has allowed us to derive the basic equation of nonrelativistic quantum physics, based on principles fundamentally different from the ideological foundations of neo-positivism. However, quantum mechanics itself, created by a galaxy of great scientists, did
not suffer from this approach. On the contrary, it has even strengthened its logical foundation.
The approach proposed in this paper also permits one to derive all the basic equations of nonrelativistic quantum physics: the Klein-Gordon equation, the Dirac equation, and so forth. The algorithms for these derivations follow steps which are analogous to the ones given in this article:
1) express the instantaneous deterministic action of the particle;
2) find the mean of this action,
3) represent all averaged terms in the integrand of the mean action by their respective probability density
32
functions ρ(х) and/or ρ(px);
4) switch all terms of the Lagrangian (the mean of the action) to a coordinate or a momentum representation;
5) determine the equation for the extrema of the resulting functionality (the mean of the action)
through methods of the calculus of variations.
The significance of this derivation of the generalized Schrödinger equation (102) is, in the author’s opinion, as follows:

The limits and conditions of the application of this equation to both the micro- and macrocosm
become clear.

There is no need to apply either Heisenberg’s uncertainty principle or de Broglie waves, since, for
the derivation of equation (102), the procedure (45) through (48) is used, and this procedure is completely analogous to the transition from the coordinate representation to the momentum one, and vice versa.

However, this procedure is obtained based only on the analysis of the properties of a stationary
stochastic process, without the involvement of the above hypotheses.
Indeed, if Louis de Broglie had not ventured a brilliant guess about the momentum of the particle
due to a certain oscillation frequency, perhaps the author of the present article might have succeeded in
establishing the procedure (45) through (48).

The quantity ħ/m (“Planck's constant” divided by the mass) is obtained through the definitions of
variance and of correlation coefficient (i.e. through their respective mean characteristics) while investigating stationary stochastic processes. Therefore, the generalized Schrödinger equation (102) does not
contain the particle’s "mass" m, which is in any case a poorly defined concept.
33
Recall that in quantum field theory, it is possible to determine, more or less correctly, the rest
mass of a particle by a structure, altogether not simple, of pseudo-symmetry breaking involving Higgs
bosons.
According to the author, "mass" is one of the "darkest" dimensional quantities in modern physics
[see Paragraph 1.7.10 in (Gaukhman 2007) and Section 7 in (Gaukhman 2008)]. There is no doubt that
the concept of "mass" should be omitted from the ultimate theory, and this article is one of the steps towards the elimination of this concept from scientific ideas about the Reality surrounding us.
By also taking into consideration the volume and the trajectory of the particle in motion, “common sense" may return to the physics of the microcosm, allowing the human mind to get back its habitual logical "footing".
I recall the words of Niels Bohr: "We all agree that our theory is crazy. We differ only in whether
it is crazy enough to have a chance of being correct."
I hope that if this article will be carefully analyzed and accepted by the scientific community,
such "delirium tremens" may finally leave the "sick folly" of quantum physics.
Of course, the ideas of the absence of distinct trajectory and sizes of elementary particles were
introduced in quantum physics due to a number of problems, such as:

The instability of the distributed (non-point) charge, as if the charge had some kind of size, then
between its near points there would be such a huge electrostatic repulsion force that the charge with
nonzero volume would immediately fly apart into small pieces, in which even greater repulsive forces
would act, and so forth.

The indivisibility of elementary particles (since, if such a particle had a distinct size, it would be
possible to cause an impact onto one of its parts in such a way that this part would separate from the rest,
but this does not happen). Also, elementary particles would then be distributed over the internal charac-
34
teristics of the space, such as stress, strain, temperature, etc. But there is no evidence of any of this in
quantum physics. It gives elementary particles (i.e. mathematical points) only those discrete characteristics as: spin, charge, magnetic moment, and so forth.

The global nonlocal nature of elementary particles, as in some experiments in which they gener-
ally manifest themselves as "spread" as a wave throughout space.
The above problems have forced the physics of the microcosm to become "sick", but the mathematical "nonsense" of quantum theorists appeared so attractive and wide-reaching that we are shocked at
how they managed to go far, wandering in complete darkness.
We merely note that many of the above problems have been solved in the frameworks of fully
geometrized theories. One of these theories is the "Algebra of signatures" (presented in the website
www.alsignat.narod.ru )
Acknowledgements
The main thesis of this article was first published in 1990 in part in series of discussions which
appeared in (Batanov 1990 / 1994). For this I thank my mentors, Drs. A. Kuznetsov and A.I. Kozlov.
I also wish to thank Drs. A.A. Rukhadze and A.M. Ignatov for their decision to publish one of the versions of this article in the journal "Engineering Physics" № 3, 2016. I also thank David Reid for his creative and highly professional approach to the present translation of this article into English.
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