ap-6-10.dvi


Acta Polytechnica Vol. 50 No. 6/2010

Non Mechanical (Mezic) Type Forces in the Foundations of
Quantum Mechanics

Č. Šimáně

Abstract

Many authors have attempted to derive the fundamental equations of quantummechanics from classical hydrodynamics.
In thepresent contributionwepresume that the continuous, electrically chargedmaterial substancemoves simultaneously
under the influence of the electric field and at the same time undergoes a diffusion process. This assumption leads to the
appearance of non-mechanical (mezic) type forces responsible for inner sources ofmatter (positive or negative), similar to
those whose existence is supposed to exist in relativistic hydrodynamics. We obtained a non-linear differential equation,
convertible by linearization to a form coinciding with the Schrödinger equation, as a condition for the establishment of
the same steady states with discrete energies.

Keywords: hydrodynamics, quantum mechanics, mezic (non-mechanical) forces.

1 Introduction

The discovery of quantum mechanics signified a rev-
olution in the physical image of our world. However,
the price paid for the general acceptance of quantum
mechanics was very high. One had to sacrifice many
fundamental principles of classicalmechanics, among
them especially its determinism, and to accept con-
troversial uncertainty relations. The constant h̄ dis-
covered by Planck together with Einstein’s relations
E = hν and �p = �kν between the corpuscular and
wave aspects of matter were the starting points for
the discovery of quantum mechanics by Schrödinger,
Louis de Broglie, Born, Heisenberg andDirac. Many
physicists tried to overspan the profound abyss be-
tween classical and quantum mechanics, proposing
variousmodels leading to the fundamental equations
of quantummechanics [1] . This work is in someway
a continuation of the author’s earlier article on this
subject [2].

2 From classical to quantum
mechanics

The Schrödinger equation for a spinless particle in
spherical S states of the hydrogenatomhasbeen cho-
sen to demonstrate that the assumption of the exis-
tence of non-mechanical (mezic) forces and diffusion
of electrically charged continuous electron matter in
the field of electric potential are sufficient to deduce
its formal equivalent from classical hydrodynamics.
Because one can hardly reconcile the continuous hy-
drodynamical system with the electrons as particles
in the planetary model of the atom, the concept of

particles is the first that had to be sacrificed. Instead
of this, one has to replace themby a cloud of charged
electronmatter around the nucleus. Further, one has
to propose some mechanism which could lead to the
existence of discrete time independent states of the
system. Inspiration comes from the paper by Nel-
son [3],whoused statisticalmechanics toderiveequa-
tions that led to discrete states in planetary atomic
systems. His basic idea was that the electron simul-
taneously executes two kinds ofmotions. One, which
is the classical kind, with velocity �v (he refers to it
as the flow velocity) in the electrostatic field of the
nucleus, and a certain type of Brownian motion re-
sulting in a motion with osmotic velocity �u obeying
the diffusion law

μ�u = −D∇μ (1a)

which leads to relations

�u = −D
∇μ
μ

, ∇μ = −D−1μ�u,

u2 = D2
(∇μ)2

μ2
, div (μ�u)= −DΔμ

(1b)

Inspired by Nelson, we constructed a heuristic La-
grangean L for the motion of a volume element δV
containing mμδV electron matter (m is the electron
rest mass, μ the distribution function normalized to
unity, eμ the electric charge density)

L =
∫

δV

μ

(
m

v2 + u2

2
− eϕ(�x)

)
dV (2)

as a sum consisting of two Lagrangeans in which the
integrand is the sum of the densities of the kinetic
potentials corresponding to the motions with uncor-
related velocities �v and �u(rot�v = rot�u = 0) and the

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Acta Polytechnica Vol. 50 No. 6/2010

density of the potential energy eμϕ. For infinitely
small δV the Lagrangean is replaceable by

L = mμ

[
v2 + u2

2
−e(ϕ − ϕ0)

]
δV (3)

From the general formula

d
dt

∂L

dq̇i
−

∂L

dqi
=0 (4)

we get the equations ofmotion of the volume element

d
dt
[μ(�v + �u)δV ]−

∇
[
μ

( (
v2 + u2

)
2

−
e
m
(ϕ − ϕ0)

)]
mδV =0,

(5)

The ϕ0 is for the moment an arbitrary constant po-
tential, which disappears in the potential gradient.
The first member on the left side of (5) is the

inertial force acting on the volume element δV by
acceleration. The second member represents the ex-
ternal force densities.[

−e
e
m

μ∇ϕ +( (
v2 + u2

)
2

−
e
m
(ϕ − ϕ0)

)
∇μ

]
mδV

(6)

The firstmember of (6) is the classical electrical force
density.

−
e
m

μ∇ϕ (7a)

The second member is interesting. If ∇μ is substi-
tuted from (1b), then this force density becomes

�uD−1μ

(
v2 + u2

2
−
e
m
(ϕ − ϕ0)

)
= �uq, (7b)

a product of the velocity �u and the sum of the densi-
ties of the kinetic and electrical potentialsmultiplied
by D−1. Setting D = h̄/(2m)with dimensionm2s−1,
q getsdimensionkgs−1 andmaybe interpretedas the
change of matter content in a volume element in the
time unit, i.e, as the density of an internal source of
matter.
Then, the equations of motion can be explicitly

written in the form[
μ
d�v
dt
+ μ
d�u
dt

δV +(�v + �u)
dμ
dt

δV +

μ(�v + �u)
(δV )
dt
+
e
m

μ∇ϕδV − �uq
]

mδV =0
(8)

The time derivative of δV in the fourth member of
(8) must be treated as the sum of the derivatives at
�v and �u constant

μ(�v + �u)
(δV )
dt
=

μ(�v + �u)[div�u(�v + �u)+div�v(�v + �u)]δV =

μ�v div�vδV + μudiv�uδV

(9)

Then the equations of motion of the unit volume el-
ement take the form

μ
d�v
dt
+ μ
d�u
dt
+�v

[(
∂μ

∂t

)
�u

+�v∇μ
]
+

μ�v div�v + �u

[(
∂μ

∂t

)
�v

+ �u∇μ
]
+

μ�udiv�u +
e
m

μ∇ϕ + �uq =0

(10)

or by putting together the third and the fourth, and
the fifth, sixth and the lastmember members in (10)

μ
d�v
dt
+�v

[(
∂μ

∂t

)
�u

+div(μ�v)

]
+
e
m

μ∇ϕ+

μ
d�u
dt
+ �u

[(
∂μ

∂t

)
�v

+div (μ�u)

]
+ �uq =0

(11)

Here again the partial timederivativesmust be taken
at �v and �u constant.

The sumof the first and the thirdmember in (11)
is the equation of motion

μ
d�v
dt
+
e
m

μ∇ϕ =0 (12a)

and the expression in the square brackets in the sec-
ond member in (11) the equation of continuity

(
∂μ

∂t

)
�u

+div(μ�v)= 0 (13a)

in the motion with the velocity �v.

The remaining members in (11) represent the
equation of motion with velocity �u, which is accord-
ing to (1) a function of μ. The last two members
in (11) can be put together, so that the equation of
motion becomes

μ
d�u
dt
+ �u

[(
∂μ

∂t

)
�v

+div(μ�u)+ q

]
=0 (12b)

Theexpression in squarebrackets in the secondmem-
ber in (12b)maybe interpretedasanequationof con-
tinuitywith internal sources ofmatter q expressedby
(7b)

(
∂μ

∂t

)
�v

+div (μ�u)−

D−1μ

(
v2 + u2

2
−
e
m
(ϕ − ϕ0)

)
=0

(13b)

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Acta Polytechnica Vol. 50 No. 6/2010

3 Steady states

From now on, we will be interested only in steady,
time independent solutions of the equations of mo-
tion and of the continuity equations. Therefore, we
shall omit all partial time derivatives.
Equations (12a) and (13a) concern the classical

motion of matter in the electrostatic field of the nu-
cleus, for which the equation of continuity (13a) re-
duces to the second member or to

�v∇μ + μdiv�v =0 (14)

The gradient of density μ may take any value be-
tween minus and plus infinity, so that equation (14)
canbe satisfiedonly for �v equal identically to zero ev-
erywhere. This of course is inconsistent with (12a),
unless another electric force density compensates the
force density in the electrostatic field of the nucleus.
The only electrical charges present that could create
the compensating electric field are the charges of the
electron cloud.
Supposing spherical symmetry of the electrical

charge, the total flux of the electrical induction
through the surface of the sphere of radius R equals
the charge within this sphere

4πR2 �E = −e4π
∫ R

ρ=0
μ(ρ)ρ2dρ (15)

Because the charges outside the sphere do not con-
tribute to the intensity, one can extend the integra-
tion on the whole space and thus, owing to normal-
ization of the distribution function, the field intensity

�E = −e
4π
4πR2

∫ ∞
ρ=0

μ(ρ)ρ2dρ =
−e
R2

(16)

The resulting electrical intensity from the positive
charge of the proton and the negative charge of the
electron cloud

En + Ee =
e

R2
−
e

R2
=0 (17)

Now, let us pay attention to the processwith osmotic
velocity �u, to which equations (12b) and (13b) per-
tain. Equation (12b) ofmotionwith osmotic velocity
has the form

�uμdiv ū + �u [div (μ�u)+ q] = 0 (18)

If the equation of continuity (13b) is valid, then the
second member in (18) equals zero and the equation
of motion reduces to

�udiv ū =
d�u
dt
=0

(
∂�u

∂t
=0

)
, (19)

in agreement with the characteristic property of the
mezic forces, which do not accelerate the material

substance, so that its spatial distribution continues
to be time independent.

After replacing in (13b) from (2) u2 by D2
(∇μ)2

μ2

and div(μ�u) by −DΔμ, we get the equation of con-
tinuity in the form

−DΔμ+D−1μ
(

v2

2
+

D2 (∇μ)2

2μ2
−e(ϕ − ϕ0)

)
=0,

(20)
which is then the condition for a time independent,
steady state of the substance in the volume element
at a given point of the cloud. Simultaneously, equa-
tion (14) must also be satisfied, which — as shown
above — is possible only for �v equal identically to
zero. Therefore, the final condition for a steady, time
independent state has the form

− DΔμ+D−1μ
(

D2(∇μ)2
2μ2

−e(ϕ − ϕ0)
)
=0 (21)

The solution μ(�x,eϕ0) of (14)depends on thevalueof
the constant eϕ0. If in all points in space the solution
fulfils (14) for the same value of eϕ0, then equation
(21) represents the condition for a steady state of the
whole cloud and may be interpreted in the following
way:
The steady, time independent state of the

cloud is reachedwhen ineachoneof itsvolume
elements the outflow (inflow) of the matter
through its surface is just compensated by the
internal positive (negative) sources of matter
evokedbynon-mechanical (mezic) type forces.
Substituting μ = R2 in the non linear equation

(21), one obtains a linear equation(
h̄
2

2m
Δ+eϕ

)
R =eϕ0R ≡ ER (22)

Equation (22), derived from a hydrodynamic model,
is in fact sufficient to bring order into most of the
experimental spectroscopic data of discrete energy
states. It does not resemble a wave equation. It
is asymmetric: on the left side we have an operator,
and on the right side a simple algebraic expression.
Replacing this expression by an operator of the form
−ih̄∂/∂t acting on a exponential function

ψ = Rexp

(
−i

E

h̄
t

)
(23)

we obtain the same equation (18), butwith operators
on both sides

Hψ = −ih̄
∂

∂t
ψ, (24)

where ψ can rightly be declared as a wave function.
The author would like to remark, that one gets

the same results if the velocity �u is taken fromthebe-
ginningaspure imaginary. Then (11) is obtainedas a

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Acta Polytechnica Vol. 50 No. 6/2010

complex equation and separating the real and imag-
inary parts one immediately gets equations (12a,b)
and (13a,b).
Before finishing this chapter, the author would

like to mention the historical paper by D. Bohm [4]
concerning the so-calledhidden variables of quantum
theory.
Bohm supposed the function ψ = Rexp(−iS/h̄),

(R and S are real functions of time and coordinates),
as the solution of the Schrödinger equation

ih̄∂ψ/∂t = −
(
h̄2/2m

)
∇2ψ + V (�x)ψ, (25)

Substituting R by μ1/2, putting
∂S

∂t
= E and �v =

∇S
m
, he obtained from (24) two equations

(
∂μ

∂t

)
+∇

(
μ
∇S
m

)
=0 (26a)

∂S

∂t
+
(∇S)2
2m

−eμϕ −
h̄2

4m

[
Δμ
μ

−
(∇μ)2
2μ2

]
=0 (26b)

Oriented on the planetary atomic model, Bohm in-
terpreted (26b) as a Jacobi-Hamilton equation for
the motion of a particle in the field of classical elec-
tric potential and in the field of a quantum poten-
tial expressed by the last member on the left side
of (26b). However, this equation,multiplied by μ/D,

with ∇S/m replacedby �v, and
μ

D

∂S

∂t
by

(
∂μ

∂t

)
�v

and

with the use of (1b) canbe transformed into equation
(13b) with an entirely different interpretation.

4 Discussion
A necessary condition for the existence of steady,
time independent spatial density distribution and
fluxes of the electronic substance in an electrically
charged material continuum is the validity of the
equation of continuity with internal sources or sinks
of matter, which compensate the outflows or inflows
ofmatter in eachof the volumeelements of the object
through its surface.
The basic prepositions necessary for the deriva-

tion of this equation of continuity reducible formally
to a form of the Schrödinger equation were that
• the electron in the bound state does not exist
as a particle and forms a cloud of continuous
electrically charged electron substance around
the nucleus, bearing the total electrical charge,
whose spatial distribution is the same as that of
the electron substance.

• there exist two kinds of uncorrelated motions of
the continuum, one with flow velocity �v and the
otherwith osmotic velocity �u, the latter obeying
the diffusion law.

• the continuous electron substance moves in the
electric field of the nucleus and in the field of

its own electrical charge and at the same time
diffuses opposite to the density gradient.

• inner sources or sinks of matter due to the ac-
tion of non-mechanical (mezic) type forces exist
in the continuous electron substance.

• in the steady time independent case, the flowve-
locity is zero and the intensity of the electrical
field of the nucleus is compensated by the elec-
tron field of the electron cloud.

• the possibility of reducing the nonlinear equa-
tions of continuity to the linear Schrödinger
equation offers a practical means for solving
them.

It is worth noting that continuous distribution of the
electron matter around the nucleus was strongly de-
fended by Schrödinger in his disputes with Born [5].

In relativistichydromechanics, the existenceof in-
ner sources of matter is attributed to the action of
forces of non-mechanical (mezic) character [6]. Non-
relativistic hydromechanics does not calculate with
their existence. As such, they have never been ob-
served, because they vanish in steady states and
one can conclude their existence only indirectly from
the fact that with their aid one can get the same
steady time independent states as from the corre-
sponding Schrödinger equation. Nevertheless, the
non-mechanical forces shouldmanifest their existence
in non-stationary processes during so-called jumps
from one state to another. The violation of the con-
tinuity equation in the steady state by external per-
turbation cancels this state and starts the transition
from a steady state to a time dependent state. This
transition process may end in reestablishment of the
initial steady state or in a transition to some new
state with different energy. Without an exact math-
ematical description of the transition process, which
is very rapidandwhichmanifests itself as a jump, one
can only predict the probability of the transitions to
the end states. The situation could change, once one
will be able to follow theoretically the transition pro-
cess in time, in which the non-mechanical forces will
play an important role.

A very serious historical difficulty was mentioned
already by Madelung [7] in his attempt to deduce a
hydrodynamical model from the Schrödinger equa-
tion, namely that one has to assume that the elec-
trically charged volume elements of the same elec-
tron cloud do not mutually interact. In the plan-
etary model, this difficulty disappears, as the inner
electric field is concentrated in the point-like particle.
We suppose that the inner electrical intensity in the
electron cloud compensates just the electrical inten-
sity in the electric field of the nucleus, which allows
the flow intensity in the discrete, time independent
state to be obtained equal to zero.

We have no idea how to describe the process by
which the continuous electron cloud takes the form

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Acta Polytechnica Vol. 50 No. 6/2010

and properties of a free particle. The dual wave —
particle character, manifested mainly in the vicinity
of the rest energy of the free particle and depending
on the type of the experiment, might be explained
as fluctuations between the continuous and particle
states of the electron, due to small perturbations of
the continuity equations provoked by weak external
fields, by their time dependence and boundary con-
ditions in various types of experiments.

5 Conclusion

The hydrodynamical equations of motion deduced
above, which are based onNewtonianmechanics and
include non-mechanical (mezic) forces combinedwith
the diffusion of the charged electronic substance in
the field of an electric potential,maybe considered in
only a limited sense equivalent to quantum mechan-
ics based on the discoveries by Schrödinger, Louis de
Broglie, Born, Heisenberg, Dirac, covering a much
broader range of experimentally observed quantum
phenomena. It is too early to draw general conclu-
sions fromthe one isolated special case treatedabove.
Nevertheless, one shouldnot omit to pay attention to
the possible role ofmezic forces in quantummechan-
ics.

References

[1] Jammer, M.: The Philosophy of Quantum Me-
chanics, J. Wiley & Sons, New York, 1974.

[2] Šimáně, Č.: Discrete states of continuous electri-
cally chargedmatter.Concepts of Physics, Vol.V,
No. 3, 2008, 499–512.

[3] Nelson, E.: Derivation of the Schrödinger equa-
tion from Newtonian mechanics, Phys. Rev. 50,
1966, 1079–1085.

[4] Bohm, D.: A suggested interpretation of the
quantum theory in terms of “hidden variables“ I,
Phys. Rev. 85, 1952, 166–179.

[5] Born, M.: The conceptual situation in Physics
andtheprospects of its futuredevelopment,Proc.
Phys. Soc., A, Vol. LXVI,1953, 501–513.

[6] Möller, C.: The theory of relativity, second edi-
tion, Clarendon Press Oxford 1972, Ruskoe iz-
danie, Moskva Atomizdat 1975, pp. 105–108;

Votruba, V.: Základy speciálńı teorie relativity,
Academia Press, Prag 1969, pp. 307–313.

[7] Madelung, E.: Quantentheorie in hydrodynamis-
cher form, Zeitschr. für Physik, 40, 1926, 166.

Prof. Dipl. Ing. Čestmı́r Šimáně, DrSc.
Phone: +420 284 819 279
E-mail: csimane@centrum.cz
Nuclear Physics Institute AS CR
Prof. Emeritus of the Czech Technical University in
Prague
Home adress:
U svobodárny 9, 190 00 Praha 9, Czech Republic

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