Transcript Document

Derivation of the Schrödinger
Equation from the Laws of
Classical Mechanics Taking
into Account the Ether
Dr.Nina Sotina
e-mail: [email protected]
The Schrödinger equation describes many observations
very well, but there are still discussions about its physical
interpretation. Currently, the probabilistic interpretation is
the most common one. Its proponents, however, have
difficulties explaining the results of experiments with
nonclassical optical effects (e.g., the two-photon
interference, teleportation of polarization of the photon,
etc.). It can be shown that the probabilistic approach
for the case of “nonclassical” optical effects can
lead to negative probabilities. It is one of the reasons
to get rid of the probabilistic interpretation of the quantum
formalism and to return to the idea of “hidden variables”
The probabilistic interpretation is also unable to
describe the behavior of quantum systems in living
matter (biomolecules). It is known that a living organism
molecules act as well-tuned mechanisms. That conflicts
with a concept of a molecule as a quantum system that is
governed by probabilistic laws of quantum mechanics.
E. Schrödinger was the first who note this conundrum.
It was one of the reasons why he was
an opponent of the probabilistic
interpretation of  - function.
E. Schrödinger believed that
 - function is associated with some
real oscillatory process in an atom.
In 1935 Einstein, Podolsky, and Rosen put forward the
issue of incompleteness of the quantum mechanics
description of physical reality and suggested the idea of
existence of «hidden variables». Later it was proven (J.Bell
and others) that «hidden variables» can be either:
1) "nonlocal" (the “nonlocality” is the existence of a connection
between spatially separated measurement devices; in essence, theory
of long-range forces), or
2) a field wherein disturbances can spread at speeds
greater than the speed of light (hence, possibility of the
existence of “superluminal” forces).
The latter was inconsistent with theory of relativity
therefore nobody tried to introduce such a field (or
medium) to get rid of the probabilistic interpretation of
the quantum formalism .
D.Bohm was an Einstein's student and the follower of
theory of relativity. He supported the idea of quantum
nonlocality and introduced the field of information in physics.
However idea of quantum nonlocality is also
not consistent with theory of relativity since
there is a implicit postulate of the “locality”
in it: the universe can be decompose correctly
into difference and separately existing
“components of reality”. To measure the
speed of light it is necessary that the receiver
and transmitter were not only separated in
space, but were also autonomous in their behaviour.
As a result more and more physicists are inclined
towards neither interpretation, which was expressed by
physicist David Mermin as “Shut up and calculate!”
I believe the way out of the crises in quantum mechanics
is in the return to the idea of «hidden variables» as a
physical field (medium, ether) wherein disturbances can
spread at speeds greater than the speed of light.
I will show below that the Schrödinger
equation can be derived from the
deterministic laws of classical mechanics.
This derivation is based on the work of
Russian scientist Chetaev N.G. (1936).
Chetaev showed an analogy between the
stable motion of a mechanical system under action of
conservative forces and quantum processes described by the
Schrödinger equation. His work was not translated in
English, and is not known outside Russia. So it is presented
here to the English speaking community for the first time.
The derivation of the time-independent
Schrödinger equation
Consider the motion of a mechanical system under the action
of conservative forces that do not depend on time t explicitly.
In this case there exists the energy integral H =T  U = ε
where U (q1 ,..., qn ) is potential energy, q1 ,...,q k are
coordinates, p1 , ... , p k are momenta, and 2T =
aij pi p j
the kinetic energy of the system.
i, j
The complete integral of the Jacobi equation


S
S
S 
 H  q1 ,..., qn ;
,...,
  0 takes the following form:
t
q1
qn 

S   t  V  q1 ,..., qn ; a1,,..., an 
S V

 pi , i  1,2,..., n
qi qi
Let’s make change of variables
   S
The following equations are correct: ψtt = ψ' ' ε 2 (  t  d / dt
 
d
),
dS

qi
 ψ   

 aij

 ψ' aij S  = ψ' ' aij S S + ψ' 
 q  q 


q
qi q j
qi
j
i
j




The latter equation can be represented as follows

L

  
where
L ψ = ψ''  aij pi p j +ψ'L  S 
q
i
or, using the energy integral
written in the form
a
ij
pi p j = 2  ε  U 
 S 
 aij

 q 
j 


 
 aij


q
j 

can be
2ε U 
L ψ  = ψtt
 ψ'L  S 
2
ε
It follows from the theory of stability that for a motion to
be stable L  S  must be zero (necessary condition of stability).
Thus for a stable motion
2ε U 
L ψ  = ψ tt
ε
2
Let us search for the solution of the latter equation in the
 i 
 i 
form
  exp  S   exp 
t   r  (1)




In this case the equation is written as the Schrödinger
equation
 ( r , t )
i

  r , t   U ( r )  r , t 
or
t
2m
2
2

2m
  r   U (r )  r     r 
(2)
Note, that E. Schrödinger also used the Jacobi equation and
the substitution (1) to derive equation (2). He found this
substitution empirically as the one which yields the RydbergRitz formula for the spectrum of the hydrogen atom.
However, Schrödinger's approach did not clearly indicate
that equation (2) contained extra solutions, because his
equation is only the necessary conditions of stability, not
sufficient.
Furthermore, it turned out that the equation (2) has more
solutions than the original problem. Probably that was the
reason why Schrödinger approach that led him to his famous
equation was not accepted by a scientific community and the
equation (2) was taken as a postulate.
Chetaev, however, noticed that the Schrödinger equation
contains other solutions besides those that are determined by
the potential energy U (q1 ,..., qn ) . He gave a method that
allows to find all the possible motions of a mechanical system
under the action of conservative forces which as a necessary
condition of stability have the Schrödinger equation (2).
Let me present now my derivation of the Schrödinger
equation based on Chetaev’s method , with the further
analysis of the solutions.
Assume that in addition to the primary forces described
by U , there are also forces described by potential W .
Represent the time-independent component of the psifunction as
  A exp  iV 
where A is some real function of coordinates. In this case
the following equations are correct:
V

q j i
 1 
1 A 



  q j A q j 


As follows from the theory of stability for a motion to be
stable the expression L[S] must be zero.
L[S] 

i, j

S
( aij
)
qi
q j

i, j
   1  1 A 
 aij 
  0

qi    q j A q j 

 
(3)
The energy integral can be represented as
2
 1  1 A   1  1 A 
  aij 



   U W

2 ij   qi A qi    q j A q j 
Conditions (3) and (4) take the form of the stationary
2
Schrödinger equation:
 
 

2

i, j
qi
 aij

q j
 U

(4)

if the additional forces Fq have structure defined by
2
  A 
A
equations
a

W 
a
p 0

i, j
ij
q j
i
2A
 q 
i, j
i
ij
q j 

Note that the forces Fq  W are different corresponding to
different solutions of the Schrödinger equation. Forces of
such type appear as an object is moving in medium (field)
as a result of object-medium interaction.
Chetaev consider forces Fq to be small . However, I
showed that they can be of the same order of magnitude as
the primary forces.
Let me demonstrate this for a hydrogen atom. For a case
of one particle the Schrödinger equation takes the form
1
mv  W  U  
2
2
where v
the speed of the particle,
v (r ) 
1
V
m
i.e. the Schrödinger equation is the law of conservation of
energy on a stable orbit.
2
Fq  W 

2m
A
A
In the hydrogen atom the electron moves along a
circular orbit with the speed
v
k
m 
k  0,1, 2, ..
 - the radius
The net force that provides centripetal acceleration
2 2
k
equals
m 3
The question arises whether this deterministic approach,
in which the characteristics such as trajectory and velocity
of an elementary particle were introduced, contradicts the
second postulate of quantum mechanics : “collapse of the
wave function”? The answer to this question was given in
the article by D. Bohm and Hiley “Measurement
Understood Through the quantum potential approach”.
(1983)
Bohm and Hiley reviewed the quantum potential approach
to quantum theory, and showed that it yields a completely
consistent account of the measurement process.
Let me remind you, that D.Bohm supported the idea of
quantum nonlocality. He has noticed, that the Schrödinger
equation can be represented as the law of conservation of
energy of stable orbits. He called the potential W
“the quantum potential”. Bohm explained his quantum
potential not as a physical quantity associated with
real forces in medium, but by introducing the field of
information in physics. I would like to make a point here
that inclusion of the “information" as a term in an equation
that contains such physical quantities as time, force and
displacement looks like a logical inconsistency.
Bohm and his followers often use an example of a ship
controlled from a shore to explain their ideas. Here the ship
controlled by a radio signal is an analogy of a quantum
particle controlled by the information field. In the case of
the ship, however, the EM signal turn on some real
forces that act on the ship (such as interaction between the
ship mechanisms and fluid). I think that Bohm‘s
information approach lacks the explanation of what is the
nature of the forces that change the particle's motion, even
if the control is done at a distance.
Moreover, D. Bohm derived his formulas directly from
the Schrödinger equation . As was mentioned above this
equation contains extra solutions , i.e. such trajectories that
do not realize in nature. This should be taken in to
consideration when one analyzes concrete solutions.
Another question is: how well does my deterministic
interpretation of the Schrödinger equation agree with the
probabilistic interpretation? The answer to this question was
in fact given by another Russian physicist V.A. Kotel'nikov
(2009) . As a starting point in his study Kotel’nikov uses the
Shrödinger equation and probabilistic interpretation of the
psi- function. He raised a question: how can an elementary
particle move according to the laws of
classical mechanics if its probabilistic
behavior is determined by the
Schrödinger equation. As result of his
mathematical derivations, he concluded
that the particle should move under the
action of two forces: a classical force, defined by the potential
U and a quantum force defined by “quantum potential” W .
The role of physics is to establish correspondence between
mathematical properties of psi-function
and physical

i

t



t
,
x

exp



  x  must change
quantities. Psi-function


continuously along the trajectory and into the space
surrounding the particle. From the latter formula we can see

that an oscillations with exactly the same frequency are at
each point of space surrounding the particle. Unfortunately
the theory does not address the nature of this process yet. We
can make some speculations only.
In my opinion the ether is similar in its properties to
superfluid 3 He . In this case the following interpretation of
the mathematical formalism of quantum mechanics for a
hydrogen atom can be suggested.
The centre of mass of an electron in a hydrogen atom
moves along a circle. Due to the Barnette’s effect, a
uniformly processing domain can appear in
superfluid, which was observed experimentally. The
frequency of precession of the domain  .
Simultaneously the electrical polarization can
appear in such a medium.



e
v