Download APPENDIX A

THEOREMS YELD VIA ARBITRARILY ALTERING THE MASS OF AN OBJECT,
IN ITS QUANTUM MECHANICAL DESCRIPTION: A FUNDAMENTAL
ARCHITECTURE, MATTER IS MADE OF
Tolga Yarman, Okan University, Akfirat, Istanbul, Turkey ([email protected]),
Savronik, Organize Sanayii Bölgesi, Eskisehir, Turkey
ABSTRACT
An arbitrary increase of rest masses entering the quantum mechanical description of an atomic or molecular
object, leads to the increase of the related total energy, and contraction of the size of the object at hand.
Furthermore, this quantum mechanical occurrence yields the invariance of the quantity [energy x mass x size2],
framing a fundamental architecture, matter is made of. On the other hand, one can check that the latter quantity
is Lorentz invariant, no matter one works on a non-relativistic or relativistic quantum mechanical basis. Thus, it
appears that, the quantum mechanical invariance of [energy x mass x size2] with respect to a hypothetical mass
change, works as the principal mechanism of the end results of the Special Theory of Relativity, were the object
in consideration brought to a uniform translational motion, or similarly the principal mechanism of the end
results of the General Theory of Relativity, were this object embedded in a gravitational field (given that, in
either case, it is question of a “mass” change, which can well be considered as an input to the quantum
mechanical description at hand). One can further show that, the occurrence we disclose holds, not only for the
gravitational field, but generally for all fields, the object at hand interacts with.
1. INTRODUCTION
It was the author’s idea that, owing to the relativistic law of conservation of energy, the
internal energy of a particle bound to a field, should be weakened, as much as the binding
energy coming into play, regardless whether the object is bound to an electric field in the
atomic world, or a gravitational field in the celestial world, or any other field, it can interact
with. Accordingly the rest mass of it, must be (for attractive fields), decreased as much. In
effect, all internal mechanisms, the particle of concern may embody, should be slown down,
when bound to a field, provided that the particle’s inner articulations in relation to each
other, are not degenerated, via the binding process, coming into play. “Rest energy”,
anyway, relativistically means, “rest mass”. If the rest energy weakens, for any reason, then,
the rest mass, contrary to the general wisdom, for which the rest mass is a universal property,
must be decreased just as much. The rest energy should be associated with the internal
dynamics, or say the internal activity of the object at hand. Thus, this energy must not really
be a universal invariant, and may very well get decreased, owing to the law of energy
conservation, broadened to embody the mass & energy equivalence of the Special Theory of
Relativity (STR), thus as much as the binding energy, the entity would display when engaged
in a given interaction. In different terms, as one furnishes the necessary amount of energy to
the entity, to get it off the bond, it is engaged in, it is essentially, the rest relativistic energy
which would pile it up. Then the rest relativistic energy, due to the relativistic law of energy
conservation, would increase, as much. This is to say that, the rest mass will increase as
much. Or conversely, the rest mass, while the process of binding, would decrease as much as
the binding energy coming into play. This means that, the energy of the internal activity of
the object of concern, when bound, decreases just as much… If so, we could very well be
able to describe the effect of the field on the object, by just decreasing its rest mass, in its
quantum mechanical description. This is the essence of our approach. In this aim, we first
sate, and prove a quantum mechanical theorem, in the case where the rest masses involved by
the quantum mechanical approach are all altered by a given arbitrary number (Section 2).
Then we show that, the quantity mass x size2 x total energy stays invariant with respect to the
a given change arbitrarily of the masses involved by the quantum mechanical description of
the object at hand; it is further rooted to the h2, the Square of the Planck Constant (Section 3).
This, interestingly delineates, a fundamental architecture, matter is made of, assuring the end
results of the STR, as well as those of the General Theory of Relativity (GTR). As an
example to the architecture in question, a whole new systematization of diatomic molecules is
presented. Finally a conclusion is drawn (Section 4).
2. A QUANTUM MECHANICAL THEOREM, AND PROOF OF IT
Herein we will first prove the following theorem, framing in fact, a fundamental matter
architecture.
Theorem: Consider a relativistic or non-relativistic quantum mechanical description of a
given object, depending on whichever, may be appropriate. This description
points to an internal dynamics, which consists in a “clock motion”, achieved in a
“clock space”, along with a “clock unit period of time”. The description excludes
“synthetic potential energies”, which may otherwise lead to incompatibilities with
the Special Theory of Relativity. The description in question is supposed to be
based on K particles, altogether. If then different masses mk0, k = 1, …, K,
involved by the quantum mechanical description of the object at rest, are all
multiplied by the arbitrary number  , the following two general results are
conjointly obtained:
a) The total energy, i.e. the eigenvalue of the quantum mechanical description in
consideration, E0 associated with the given clock’s motion of the object is
increased as much, or the same, the unit period of time T0, of the motion
associated with this energy, is decreased as much.
b) The characteristic length, or the clock space size R 0 to be associated with the
given clock’s motion, contracts as much.
In mathematical words this is:
[(mk0, k = 1, …, K)  (  mk0, k = 1, …, K)]  ( E 0   E 0 ) or ( T0 
(R 0 
T0

), and
R0
)].

In this theorem, we come to introduce somewhat new concepts, such as internal dynamics,
clock motion, clock space, clock space size, clock unit period of time, or straight, period of
time. In the introduction above, we already used the denominations of internal mechanism,
internal action, or straight action, internal activity, internal energy, or the same, energy of the
internal activity. And we will soon add other, still somewhat new concepts, to our glossary,
such as, overall internal dynamics, or global internal dynamics, overall periodic
phenomenon, internal motion, sub motion, clock labor, clock mass, clock energy, sub activity,
thus partial energy associated with a sub activity. Before we prove our theorem, let us briefly
explain and define these concepts.
Louis de Broglie, in his doctorate thesis, wrote h 0  m 0 c 2 for an object of rest mass m0;1
here h is the Planck Constant, c the velocity of light in empty space, and  0 is the frequency
of the electromagnetic radiation, that one would tap if the mass m0 in question, were entirely
annihilated.
What de Broglie wrote, is in fact nothing else, but the equality of the energy expressed
Planck-wise,2 and the energy expressed Einstein-wise.3 What is important from our point of
view, is that by doing so, de Broglie defines a periodic phenomenon to be associated with the
rest mass m0, in its entirety. The frequency of oscillation of this phenomenon, becomes  0 ,
i.e. that of the electromagnetic radiation, that one would tap, if the mass m0 in question, were
totally annihilated..
It is thus question of an “overall internal dynamics”, the way we like to coin. It is periodic.
Thus we like to call it, the “overall periodic phenomenon” the object delineates, internally.
(This is the content of de Broglie’s original insight.) Inside of this whole thing, though, there
may be several sub motions, the object would depict.
Consider for instance a diatomic molecule. It rotates. It vibrates. The electrons making the
molecular bond, can well be considered to be involved with the vibration motion, etc. Inner
shell electrons, on the other hand, may be considered to display different actions. We call,
each of these motions, internal motion. Any of these, generally, keep on repeating itself
regularly.
Thus, there is something going on, within any object, no matter how elementary the object
may be. We call it, internal motion. It displays an internal mechanism. In most cases there are
many things, going on, inside any object, along with different internal mechanisms. Each of
these, is to be associated with a particular “internal dynamics”, next to the global one
defined by de Broglie, himself, based on the pair m0 and  0 . Any internal dynamics thus
consists in a given motion. Any such motion can be considered as a clock motion. Thus the
internal dynamics works as a clock. The clock motion is generally an oscillatory regular
phenomenon. The clock thus displays, a regular clock labor. It is an internal activity. This
takes place in a clock space. The size of this space is the clock space size, or in short, just the
clock size. We can call it, as well, the clock characteristic length, or just characteristic
length.
The clock labor is carried by the clock mass. The clock mass may not be a simple mass. For
instance for a Bohr atom, where the nucleus is considered at rest, the clock labor is the
rotation of the electron around the nucleus, and the clock mass is the electron mass. If the
motion of the nucleus is not neglected, then the clock mass becomes the reduced mass of the
mass of the electron and the mass of the nucleus. The clock size R 0 in the Bohr Atom, is the
radius of the atom. For a diatomic molecule, it is the internuclear distance, etc. R 0 , in fact,
may be just any length one may pick up, within the framework of the object at hand, and the
above theorem, would still be valid. The clock labor delineates a clock unit period of time, or
just the clock period.
Each internal dynamics delineates a total energy. This is the eigenvalue E0 to be yeld by the
related quantum mechanical description. We can call it internal energy, or clock energy, or
energy of the internal activity. Again the total rest mass of the object represents an overall
internal activity, through the de Broglie equality, h 0  m 0 c 2 . Under the umbrella of this,
one can depict several sub activities. The internal activity of a radioactive nucleus for
instance, is radioactivity. With each of the sub activities in question, one can associate a
partial total energy E0, to be obtained from the related quantum mechanical description, if
nature allowed, a practical separation of different activities from each other, at least for a
practical mathematical analysis.
Proof of the First Part of the Theorem: [(mk0, k = 1, …, K)  (  mk0, k = 1, …, K)] 
[ E0   E0 ]
For our purpose, for simplicity, without though any loss of generality, we consider the (time
independent) Schrödinger Equation, i.e. with the familiar notation, written for an atomistic or
a molecular object composed of J nuclei, of respective masses mj0, j = 1, …, J, and I electrons
(altogether), of (the same) mass mi0, i = 1, ..., I:
2
2
2
2
2

Z e
Z Z e 
   h  2j   h  i2   j0   e   j0 j'0   0 (r 0 )
2
 j 8 2 m
rij0
r jj'0 
i, j
i 8 m i 0
i ,i ' rii '0
j, j'
j0

= E 0  0 (r 0 ) .
(1)
E0 is the eigenvalue, and  0 (r 0 ) the related eigenfunction; e is the charge of the electron, h
the Planck Constant; Zj0 is the atomic number of the jth nucleus; rij0 is the distance between
the ith and the jth particles.
Eq.(1), already represents a given complexity. Instead, we could have considered straight a
much more general quantum mechanical description, with respect to a given number of
unspecified particles, without making any distinction between these, even those which would
be identical. But then the picture we would have to paint, would have remained too abstract,
and the derivations that would result, too cumbersome to follow. This is why we avoid a more
general quantum mechanical description, to start with.
On the other hand, in any case, any potential energy term to be input to such a description
must be compatible with the STR. Otherwise, even a relativistic Dirac description would
furnish results which do not consist in Lorentz invariant forms.
Thus, multiply all electron masses mi0 (i = 1, ..., I), as well as all nuclei masses mj0
(j = 1, …, J), appearing in Eq.(1), by  ; the eigenfunction and the related eigenvalue will
accordingly be altered to become, respectively,  new (r 0 ) and E:
2
2
2

Z j0 e
Z j0 Z j' 0 e 
h2
e2
2
2
  h
  new (r 0 )






i 8 2 m i 


j
 j 8 2 m

r
r
r
i
,
j
i
,
i
'
j
,
j
'
ij
0
ii
'
0
jj
'
0
i
0
j
0


=E  new (r 0 ) .
(2)
This is the same as
2
2
2
2
2

Z e
Z Z e
   h  2j   h  i2   j0   e   j0 j'0
2
 j 8 2 m
rij0
r jj'0
i, j
i 8 m i 0
j, j'
i ,i ' rii'0
j0




=  E  new (r 0 ) .



  new (r 0 )


(3)
At this stage it is interesting to note that, through the manipulations of concern, the products
involving electric charges were left free of  . Let now
r 0  r  r 0 ,
(4)
(r )   new (r 0 ) .
(5)
(r0 ) (r ) u

; u 0  x 0 , y 0 , z 0 ; u  x, y, z ;
u 0
u u 0
(6)
together with
Since
we have
(r0 )
(r )

.
u 0
u
(7)
Eq.(3) thus becomes


2
2
2

Z j0 e
Z j0 Z j' 0 e 
h
e
h
2
2
2
2
  2
 j   2
 i  


(r)
 j 8 m
rij0
r jj'0 
i, j
i 8 m i 0
j, j'
i ,i ' rii'0
j0






2
2
=  E(r) .
(8)
Dividing by  2 , and using Eq. (4), this yields
2
2
2
2
2

Z e
Z Z e 
   h  2j   h  i2   j0   e   j0 j'0  (r)
2
 j 8 2 m

rij
r jj'
i, j
i 8 m i 0
i ,i ' rii '
j, j'
j0


E
= (r ) .

In comparison with Eq.(1), we can deduce at once, that
E
 E 0  E  E 0 (c.q.f.d.) .

(9)
(10)
Soon, we will demonstrate that, a period of time associated with the total energy E 0 ,
conjointly, is reduced just as much.
Thus, we have come to achieve the demonstration of the first part of the above theorem.
One can question the following. Isn’t quite normal that one obtains the above result with
respect to an arbitrary increase of mass, since mass and energy are anyway equivalent? Yes
of course. But conjointly, we have to recall the following.
i) First of all, let us emphasize that, such a change, to a first strike, may seem to be
hypothetical, and has (to our knowledge), so far (besides our own work), never been
considered. ii) In our approach, though, the rest mass change, has well a meaning, as
explained in the Introduction Section, above. It is that, either the STR, or the GTR, well
depict mass changes, and via the derivation presented herein, we aim to be able to explain the
end results of these theories, via mechanisms built on mere quantum mechanics. iii) In our
approach, even with regards to fields other than gravitation with the object at hand, may
interact, we expect to pin down the same mechanisms. iv) It is of course comforting to have
landed to an expected result (i.e. an increase in the rest mass, normally leads to an increase
of the related energy). v) But we have to recall that, the above derivation is achieved within
the frame of the non-relativistic quantum mechanics. And it was not obvious that, based on
even non-relativistic quantum mechanics, one would land at a relativistic relationship, as to
were mass increased, energy will increase by the same amount. vi) What is in any case not
known so far, is that, how size of an object would get affected accordingly. And this will be
revealed right below.
Proof of the Second Part of the Theorem: [(mk0, k = 1, …, K)  (  mk0, k = 1, …, K)] 
[ R 0  R 0 /γ ]
Next we focus on a size of interest R 0 (i.e. as we just pointed out, the size of an atom,
anyway we would like to define it, or the average internuclear distance in a diatomic
molecule, or whatever, through which the internal dynamics, or the same the given clock
motion of the object takes place), to be associated with the wave-like object at hand. R 0 shall
be determined based on the solution of Eq.(1). Following the mass perturbation, R 0 becomes
R 0 new , and this latter shall be found based on the solution of Eq.(2).
According to Eq.(4), R 0 new is transformed into R, so that R= R 0 new . (Note that according to
this equation, any distance, say r0 we would consider, becoming ronew due to the mass
change, is transformed into r, so that r   ronew .
Thus the derivation presented herein, in fact holds for any distance, thence also for a given
specific distance R 0 we would pick up.)
R is to be determined as the solution of Eq.(9). But since this equation is identical to Eq.(1)
[along with Eq.(10)], the solution of Eq.(9) in regards to R, is the original size of interest, i.e.
R 0 . Hence
R 0 new  R 0 ,
(11)
or the same
R
R 0 new  0 (c.q.f.d.) .
(12)

This ends the demonstration of the second part of the above theorem.
Proof of [ E0   E0 ]  [ T0  T0 /γ ]
Before we proceed, we like to draw Table 1, where we display for different cases (i.e. nonrelativistic and relativistic particle in the box, hydrogen atom, rotation of a diatomic
molecule, alpha disintegration of a radioactive nucleus), the magnitude of the eigenvalue E0
of the related quantum mechanical description, together with the period of time one can
associate with this energy. For different, in fact well known expressions used therein, for
example, the References by Davis, Eisberg and Evans can be consulted.4,5, 6
Table 1 Total Energy, and Period of time of different wave-like objects, as the solution of
basically, Schrodinger Equation, thus showing the form of {mass x size2}
Case
Magnitude f the
Total Energy (E0)
(Eigenvalue)
Period of Time of
Oscillation, or
Rotation, or
Disintegration
Half Life (T0)
Definitions
m 0 : particle’s rest mass
Non-relativistic
particle in the
(infinitely high) box
Relativistic
particle in the box
(Dirac Solution)
Hydrogen Atom
(Schrodinger
Solution)
Rotation of a
Diatomic Molecule
Alpha
disintegration
(Half life)
2
2
2
n h
8{m 0 L20}
4{m 0 L 0}
n 2h 2
8{m L20}
4{m 0 L20}
nh
L 0 : box width
n : quantum number
nh
m0 : particle’s relativistic
mass
L 0 : box width
n : quantum number
 0 : reduced mass of
2
2
n h
8 { 0 R 02}
2
(j  1) 2 h 2
8 2{M 0 r02}
n 2h 2
8{m 0 L20}
(This is the total
energy of alpha, in the
“box” of width L0 ,
before it tunnels out)
4
electron and proton
2
2
{ 0 R 0}
nh
R0 :
r0 : internuclear distance
M 0 : reduced mass of
atoms
j : quantum number
4 2{M 0 r02}
(j  1)h
2
2 ln 2{m 0 R 0}e
atomic radius
γ
h
T0  ln2/  0 ,
 0  po / R 0
 h /( 2mo R 02 )
m 0 : alpha mass
R 0  L0 / 2 ,
nuclear radius

e : barrier transmission
coefficient
p 0 : momentum of alpha
in the box
The period of time T0 of the given motion, is calculated based on E0, were the potential
energy zero, just the way it is, for most of the cases we have considered herein. In the nonrelativistic case, for instance, E0=(1/2)m0v02, m0 is the mass of the particle at hand, and v0 is
the velocity, it draws in the motion of concern.
Knowing the space size, in which the motion of concern takes place, one can then tap the
period of time delineated by the motion. For the hydrogen atom we assumed that the electron
rotates around the nucleus. The kinetic energy of it, is then half of E0. What is mainly to be
noted in Table 1, is that the total energy is architectured in such a way that, it is inversely
proportional to the cast mass x size2, the proportionality constant being rooted to the square of
the Planck Constant, and the related period of time is architectured in such a way that, it is
proportional to the same quantity mass x size2, the proportionality constant being rooted to the
inverse of the Planck Constant. We will elaborate on this below.
Anyway, a glance of eye at Table 1, at once makes it clear that, there is a simple relationship
between E 0 and T0, which is
ah
E0 
,
(13)
T0
where a is a coefficient, insuring the equality.
For instance, for the non-relativistic particle in the box, a=1/2; we assume it can be
generalized to most complex cases. Now, we can go back to the quantum mechanical
description of the object at hand. Following the multiplication of the rest masses involved by
this description, by the arbitrary factor  , the total energy E 0 has increased by the factor 
[cf. Eq.(10)]. But Eq.(13) tells us that E0 and T0 are inversely proportional to each other.
Thence we have
( E 0   E 0 )  ( T0 
T0

)
(c.q.f.d.) .
(14)
3. A FUNDAMENTAL ARCHITECTURE, ATOMIC AND MOLECULAR OBJECTS
DELINEATE: E 0 M 0 R 02 ~ h2
The theorem we have just proven, says that, if an object ever experiences, a rest mass
decrease, then, its total energy weakens as much, yielding the stretching of the period of time
of its internal motion (which should be considered quite understandable).
Let us now define a quantity M0, which we call “clock mass”; it is a mass whose motion
makes the internal dynamics of the object; it is manufactured based on different masses
embodied by the object at hand; thus multiplying all of these masses by γ, alters M0 just as
much. Let us stress that, the clock mass, is the reduced mass of the proton and the electron, in
the case of the hydrogen atom (cf. Table 1). It is the rest mass of the particle, for the particle
in the box. It is the relativistic mass of the particle for the relativistic particle in the box, etc. It
is in short, the mass entering in the expression of the internal motion’s period of time, or that
of the total energy, associated with the quantum mechanical description of concern, were this
reduced to a single mass description, such as in effect, the case for the hydrogen atom, or the
description of the rotation of a diatomic molecule (cf. Table 1).
8
Eqs. (10) and (12) (obtained with regards to an overall mass change of the object at hand),
immediately yield the invariance of the quantity E 0 M 0R 02 . We call this, the “quantum
mechanical invariance” of E 0 M 0R 02 with respect to a mass change, arbitrarily input to the
description of the object at hand.
E 0 M 0 R 02 is A Quantum Mechanical Invariant, Whether the Description at Hand, is
Non-Relativistic or Relativistic
We have to note that whether the description at hand is relativistic or non-relativistic; the
quantum mechanical invariance of E 0 M 0R 02 , with respect to an arbitrary change in the rest
masses that enter into the quantum mechanical description; anyway, holds. This is in any case,
we have proven. As we will elaborate further on, right below, it is that the quantum
mechanical description may very well not be relativistic, but the solution of it, still remains
Lorentz invariant. And this is what insures the quantum mechanical invariance of E 0 M 0R 02 ,
with respect to a rest mass variation. Or vice versa, the Lorentz invariance of E 0 M 0R 02 , for
given entity makes that, in the quantum mechanical description of this entity, an arbitrary
change in rest masses, leads to a change in its size, and its total energy, but leaving the
product E 0 M 0R 02 unchanged. For instance, the product E 0 M 0R 02 reads as 8E 0m 0 L20 , for the
relativistic particle in the box, where m0 is the relativistic mass of the particle, the box being,
all together at rest. And of course, the solution of the Dirac Equation for the particle in the
box, i.e. 8E 0m 0 L20  n 2 h 2 (cf. Table 1), is relativistically invariant. But solutions of
Schrodinger Equation too, are relativistically invariant. This can be right away checked on the
cases listed in Table 1. As an example, for the non-relativistic particle in the box, we read
8E 0 m 0 L20  n 2 h 2 , where m0 is the rest mass of the particle. Since h (the Planck Constant) is
Lorentz invariant, then obviously the product E 0 m 0 L20 well remains Lorentz invariant, thus
even for the non-relativistic particle in the box, the box still being, all together at rest. For a
quantum mechanical solution to be relativistically invariant, the only requirement, as
mentioned, is that the potential energy terms input to the relativistic or non-relativistic
description at hand, are compatible with the STR. Coulomb potential is; so is Yukawa
potential. A discussion about the subject is provided in a recent publication, where however
the mathematical description of the subject is kept at a simplest level.7
E 0 M 0 R 02 is Strapped to the Square of the Planck Constant
In any case, it is remarkable that the quantity E 0 M 0R 02 (next to its quantum mechanical
invariance with respect to an arbitrary change in the rest masses entering in the quantum
mechanical description in question), turns out to be Lorentz invariant as well (were the object
brought into a uniform translational motion).
We can further show that, the quantity E 0 M 0R 02 for any object, is necessarily strapped to the
square of the Planck Constant, h2. First of all, this becomes apparent, for all of the cases listed
in Table 1. The thing is, the cases listed in this table, are more or less simple. Our claim is
though, most general.
9
Thus no matter how complex may be the object at hand, the quantity E 0 M 0R 02 is always
strapped to h2, being proportional to it, through generally a complex, dimensionless, and
relativistically invariant quantity, which is somewhat a characteristic of the bond
configuration of the elements making up the object at hand. For the cases presented in Table
1, the proportionality constant appears as the square of a quantum multiplier x the inverse of
an integer x (occasionally) the inverse of  2 .
In more complex cases, the same occurs, the only difference being the erection of an
additional Lorentz invariant coefficient, next to the rest.
This outcome can further be grasped through the following way. i) E 0 M 0R 02 is a Lorentz
invariant quantity. ii) Essentially, the three characteristic quantities E 0 , M 0 , and R 0 are
related to each other, through the quantum mechanical description at hand, inducing the
presence of a fundamental relationship in between them. iii) All the more that the quantity
E 0 M 0R 02 is an invariant, with respect to an arbitrary change in rest masses to be input to the
quantum mechanical description in consideration. iv) It is thus question of both a quantum
mechanical invariance with respect to an arbitrary rest mass change, along with a conjoint
Lorentz invariance were the object brought to a uniform translational motion. v) This cannot
be expected to be a coincidence. vi) Thus, the quantity E 0 M 0R 02 must be somewhat rooted to
a quantity, made of a universal constant, insuring both of the invariances. vii) Such a constant
should naturally be displayed, the quantum mechanical description in consideration. viii) Such
a description is well acceptable, if the potential energy input to it, is zero (see the footnote,
below). ix) All these considerations make that the Lorentz invariant universal constant, we
have come to search is made of h, and nothing but h. x) The examination of Table 1, makes it
further obvious that, any quantum mechanical product E 0 M 0R 02 is necessarily strapped to h2,
innately encompassed by the related quantum mechanical description.
Thence:
E 0 M 0R 02 ~ h2 ,
(quantum mechanical invariance yeld by the change of mass
input to the quantum mechanical description in consideration,
strapped to the square of the Planck Constant
(15)
We call the content drawn by this relationship the UMA (Universal Matter Architecture)
Cast. It discloses already many structural properties, otherwise left obscure since very many
decades.8,9,10,11,12,13 Note primarily that, what we state along with Eq.(15), is not at all, the
result of a mere dimension analysis; in effect E 0 M 0R 02 would not be invariant with regards to
a rest mass change, if the description of the object in consideration were made of potential
energy terms incompatible with the STR, though of course, dimension-wise there would still
be no problem.*
*
Consider for instance a particle in a potential well of depth U0 , width L0 , and infinitely long plateau. The
eigenvalue of the Schrodinger Equation (along with the definitions given in Table 1, with regards to a particle
in the box), is (cf. Reference 5),
2 h 2
(i)
E 0 n  2n 0 2 ;
2 m 0 L 0
the constant ε n 0 appearing in front of h is to be determined from the transcendental equations
10
Moreover, the product of any three quantities energy, mass, and square of a given size picked
up from the body of the object at hand, would still lead to an invariant quantity, with regards
to an arbitrary change in rest masses input to the quantum mechanical description, at hand;
but such an arbitrary product would not mean anything particular, and would not evidently be
strapped to h2. Nevertheless, once the quantities energy, mass, and size we pick, are anyhow
to be interrelated, such as the case of E 0 , M 0 and R 0 ; the STR, stringently imposes a unique
architecture between E 0 , M 0 and R 0 (and this, already at rest), which is precisely the
proportionality of E 0 M 0R 02 , to the Lorentz invariant universal constant, h 2 . The given spatial
dependency of the forces bringing together, the elements of the object at hand, already
constitutes a major ingredient of it.
E 0 M 0 R 02 ~h2 Insures the End Results of the STR, as well as those of the GTR
Thus, in order to insure the end results of the STR, when brought to a uniform, translational
motion, or the end results of the GTR when embedded in a gravitational field, or any other
field interacting with, the object at hand, already at rest, must be structured in just a given
way.
 n 0 tan  n 0  2  2 m 0 U 0 L20 / h 2   2n 0 ,
 n 0 cot n 0  2 m0 U L / h  
2
2
0 0
2
2
n0
.
If say, 2 2 m 0 U 0 L20 / h 2 is chosen to be 16, then one, based on Eq.(ii), gets
E0  0.0098U0 ,
(ii)
(iii)
(iv)
and
E0  0.81U0 .
On the other hand one, based on Eq.(iii), gets
E0  0.37U0 .
(v)
(vi)
Clearly, ε n 0 acts as a quantum multiplier (which is however not an integer). The thing is, this multiplier
depends on the value of 2 2 m 0 U 0 L20 / h 2 , and from our stand point, particularly, on the mass m0 of the
particle. Thus, if the mass is arbitrarily varied, then, because we have 2 2 E 0 n m 0 L20   2n 0 h 2 [cf. Eq.(i)], the
combination E 0 n m 0 L20 will not remain invariant. And why? The physical reason is that, the potential well of
depth U0 , width L0 , and infinitely long plateau is not compatible with the STR. That is, it cannot be obtained
for instance as such, from the Klein Gordon Equation (cf. Reference 5), which is written on purely relativistic
considerations, thus well yielding the Coulomb Potential, for zero rest mass, and the Yukawa Potential for a
finite mass of the particle. The mathematical reason for the incompatibility coming into play, is simply that, a
constant value for the plateau of the potential energy injected into the Schrodinger Equation, or even a Dirac
Equation, does not land itself, to the scaling we proposed via Eq.(4) of the text, to be implemented in Eq.(3),
and as a consequence, one can not obtain Eq.(9) (requiring a 1/r dependency for the potential energy terms, in
order to yield Eq.(10) of the text. Note that, a “zero potential energy” would not matter, in that aim, and is
well acceptable. The conclusion is that, the quantum mechanical invariance of E 0 m 0R 02 with respect to an
arbitrary change, of rest masses in the quantum mechanical description of concern, even on a relativistic basis
does not generally hold, for artificial potentials (i.e. potentials not existing in nature), such as the one we have
considered above. Then, it would not be true that one can land at Eq.(15) of the text (representing the quantum
mechanical invariance of the quantity with respect to mass change in the related quantum mechanical
description, next to the Lorentz invariance of it), on simple dimension analysis considerations, for as we have
just shown, herein, in an inappropriate case, the quantity 2 2 E 0 n m 0 L20   2n 0 h 2 [cf. Eq.(i)], becomes a mass
dependent quantity, through the mass dependency of the (dimensionless) quantum multiplier ε n 0 [cf. Eqs. (ii)
and (iii)], and is not an invariant with respect to an arbitrary rest mass change, at all.
11
Or, the other way around, because the object, already at rest, is structured in just that given
way, it well yields the end results of the STR, when brought to a uniform, translational
motion, or the end results of the (GTR), when embedded in a gravitational field, or possibly
any other field apt to interact with the object. Note that, the same holds for any field the object
at hand can interact with.
This idea is elaborated on, in our recent article (cf. Reference 7), in trying to reveal the
mystery behind the Galilean Principle of Relativity,14,15 which states that an observer inside a
of a vehicle engaged in a uniform translational motion, has no way to detect, the velocity he
cruises with, as regards to, say, distant stars, by performing any experiment inside of his
vehicle. Why? Because matter is built in such a way that, the product E 0 M 0R 02 remains as an
invariant with respect to a rest mass change one may arbitrarily input the quantum mechanical
description of the object at hand. The quantity E 0 M 0R 02 is further rooted to h2. This is the
basis of its Lorentz invariance, thus more fundamentally the basis of the Galilean Principle of
Relativity, which is in fact, the principal ingredient of the STR16,17 as discussed in
Reference 7.
Thence the property E 0 M 0R 02 ~ h2 [Eq.(15)], works as the internal mechanism of the STR,
were the object brought to a uniform translational motion. Or conversely, because the product
E 0 M 0R 02 is a Lorentz invariant quantity, in order to insure the end results of the STR, were
the object brought to a uniform translational motion, the object must be, and this, already at
rest, structured, in accordance with the relationship E 0 M 0R 02 ~ h2 .
Thereby, the mass increase we introduced, to arrive at the above theorem, may very well not
be arbitrary, and this is indeed, what one experiences for instance, when a clock is removed
out of a gravitational field; its rest mass, following our approach, as required by the law of
conservation of energy, broadened to embody the mass & energy equivalence of the STR,
should be increased as much as the binding energy, the object displays vis-à-vis the host
celestial body of concern, just like the mass of the hydrogen atom is increased, as the electron
is removed away, from its orbit around the proton.18,19
The unit time displayed by the internal dynamics of the object at hand, according to the above
theorem, should then be altered, as much.
A rest mass change is anyway predicted on the basis of the GTR.20 According to our
approach, though, in contrast with the GTR, say, the gravitational red shift,21 is not a
phenomenon imposed by the curvature of space induced by the presence of host body of
concern, but is just a quantum mechanical phenomenon. Indeed, according to our approach,
the rest mass of an object bound to a celestial body, decreases as much as the binding energy
coming into play, and this, when injected into the quantum mechanical description of the
object, at hand, leads to the decrease of the object’s rest relativistic energy, and accordingly
the total energy of any quantum mechanical description associated with a given internal
dynamics it displays, just as much, thus the gravitational red shift. In any case, it must require
quite a captivation, not to consider a light phenomenon such as red shift, in mere terms of
quantum mechanics; amazingly though the history, now for many very decades, gave birth to
the unexpected.
12
E 0 M 0 R 02 ~h2 Insures the End Results of Any Process Through Which the Object
Interacts with any Given Field
Furthermore, in our approach, what occurs in a gravitational field, would also occur in an
electric field, say regarding an ionized quantum mechanical clock,22 in other terms our
approach does not have to consider any privileged field.23 Thus, if a muon is bound to a
proton, its half life should quantum mechanically stretch, as much as its binding energy. This
happens, to our knowledge, something totally overlooked. And it is worth to state Eq.(15), as
our next theorem. In other words, just like a celestial body alters the metric nearby, an electric
charge, and in particular an atomic nucleus too, alter the metric in the vicinity, in just the
same manner.
Theorem: The quantity [clock energy x clock mass x (clock space size)2], composed out of
any appropriate quantum mechanical description of a given object, turns out to
be a universal Lorentz invariant quantity strapped to h2, the Square of the Planck
Constant, and this, vis-a-vis any field the object interacts with, portraying a
unique matter architecture, delineated by atomic and molecular objects,
configured by means of the known structure of Coulomb force, reigning between
electric charges.
One can easily check that Eqs. (10) and (12), could be, as expected, obtained with respect to a
relativistic quantum mechanical description. Furthermore, one can check that these results
still hold if nuclear potentials are input to the quantum mechanical description in
consideration, instead of Coulomb potentials, when appropriate. Thus we can conjecture that
the above theorem holds universally, for all known objects.
Empiric Relationship Known, Since 1925
Note that, Eq.(15) for a constant clock mass, reads
E 02R 02 ~ h 2  Constant .
(16)
This via Eq.(13) yields the cast
T01R 02 ~ Constant .
(17)
This is nothing else, but one of the most known empirical relationships framed already in
1925, with regards to the excited levels of a diatomic molecule,24,25,26; it is now brought to us,
via our principal finding, presented herein, i.e. Eq.(15); T0 is then, the classical vibrational
period, and R0 internuclear distance, at the given level. The peculiarities of this relationship,
along with our disclosure were clarified in Reference 10.
New Systematization of Diatomic Molecules as a Result of our Disclosure
Along the line we pursued; we had drawn, Figures 1 – 7, for the entire body of known
diatomic molecules. Amongst other things, they show how beautifully, the lowest classical
vibrational period of time (T0) is architectured versus the clock mass (M0me)(1/2) (composed of
the product of, the nuclei reduced mass M0, by the electron mass me), and the internuclear
distance (r0), to provide us with the Lorentz invariant interrelation i.e. (cf. References 8
and 9),
13
T0 ~ Μ 0 m e r02 .
(18)
Note that the electron mass me is anyway, a constant.
In these figures molecules are grouped, with regards to their chemical, i.e. electronic
properties. For instance alkali molecules compose the first group. They are made of, just one
covalent bond, involving the outmost single electron of each the two atoms making the
molecule of concern. The proportionality constant becomes 4 2 g / h r0 / r00 (and we will
soon elaborate on it); here r00 is the internuclear distance of the molecule bearing the lowest
vibrational period of time of the group of molecules, we picked. We associate accordingly, the
dimensionless coefficient g, with the electronic structure of the chemical family of concern.
We indeed supposed that g should remain constant, for diatomic molecules for which the
bond electronic configuration is similar. And this is in fact, how we grouped the molecules
sketched in the attached figures. Note further that the ratio r0 / r00 corresponds to the product
of the quantum numbers, each associated with the electrons (of the diatomic molecule at
hand), taking place in the bond of the molecule (References 9 and 12); we have borrowed
data, from Herzberg.27
Our Eq.(15) may quickly shed light, to this somewhat cumbersome picture, and we will now
present a novel derivation of Eq.(18), along with the proportionality constant it embodies, to
finally arrive at our previous finding (cf. References 8 and 9), i.e.
4 2
(19)
T0 
gΜ 0 m e r02 .
h r0 / r00
Quick Derivation of the Vibrational Period Relationship For a Diatomic Molecule
Thus, we consider the vibration phenomenon of a diatomic molecule. This phenomenon
involves the vibration of the nuclei, within the electronic cloud of the electrons making up the
bond of the molecule. It is known since early times, that, the quantum mechanical description
of, on the one hand, the electronic motion of the molecule, and on the other hand, that of the
nuclei motion, can be separated from each other, out of the overall description, taking into
account the two, all together.28
Then we have two distinct phenomena. The first one is the vibration of the nuclei. The second
one the vibration of the bond electrons. The two should be carried with the same vibrational
period T0, of course, and in the same space size, i.e. the given internuclear distance, r0. The
clock mass associated with the first phenomenon, is the reduced mass of the nuclei M0 (cf.
Table 1). The clock mass associated with the second phenomenon, similarly can be pinned
down to be the reduced mass of the bond electrons. This is then nothing else, but me/2 for a
single-bond, just me for a double-bond, 3me /2 for a triple-bond. In any case it is proportional
to me. Now we propose to apply Eq.(15), twice, i.e. first for the nuclei motion of the molecule,
and second, for the electronic motion of it, with regards to its vibration:
E 0 Rot M 0R 02 ~ h2 ,
(20)
and
E 0 e m e r02 ~ h2 ;
(21)
14
here E 0 Rot is the eigenvalue of the quantum mechanical description pertaining to just the
nuclei motion of the molecule, and E 0e the eigenvalue of the quantum mechanical description
pertaining to just the electronic motion, of the bond. Via Eq.(13) one can replace these
quantities with the corresponding vibrational period, which is, as we pointed out, the same,
for both descriptions. Thus we have
1
T0 ~ Μ 0 r02 ,
(22)
h
and
1
T0 ~ m e r02 .
(23)
h
We can multiply these latter relationships side by side, and take the square root of the
outcome, to obtain Eq.(18):
1
T0 ~
Μ 0 m e r02 ,
(24)
h
where we have included the inverse of the Planck Constant, in the proportionality constant.
So we already have another quick, but efficient application of our disclosure, i.e. Eq.(15). We
know that this form is well Lorentz invariant. It embodies the Planck Constant, only, as the
Lorentz universal invariant Constant (we discussed above). One can compose many other
Lorentz invariant forms, as can be checked on the simplest entity, i.e. the Bohr Atom (BA). A
first group, which we will call G1, interrelating, two by two, clock mass (the electron mass in
the BA), clock period time (the period of time of rotation of the electron around the nucleus in
the BA), and clock space size (the atom radius in the BA), are based on both the Planck
Constant, and the (two by two products of) electric charges (were these involved). Note that
electric charges too, are Lorentz invariant. Eqs. (22) and (23) in fact, can thus theoretically be
obtained, as a compact form, via the elimination of electric charges, out of the three
corresponding G1 relationships.
The thing is, the appearance of h, in a relationship, at once evokes, the presence of a quantum
multiplier to take place next to h, in Eq.(23). This multiplier is in complex cases, because of
quantum defects, not an integer, and it is a priori not easy to determine it. We studied this
problem in References 7-12. Fortunately we can find a short cut to the result, and this is what
we will do right below. Indeed one can eliminate not only electric charges out of the three G1
relationships (relating two by two the clock mass, the clock period of time, and the clock size),
we have just mentioned, but also, the Planck Constant, thus keeping the electric charges,
instead. It will read as
1
T02 ~ 2 M 0 m e r03 ,
(25)
e
where again the proportionality constant is well dimensionless; we can rearrange this, to write
r2
1
(26)
T0 ~
M0 me 0 .
e
r0
This relationship does not involve h. And at the same time, it certainly gives us a hint about
the quantum multiplier that will take place next to h, in Eq.(24). All we have to do is to set
equal the right hand sides of Eqs. (24) and (26), to each other, via the introduction of an
appropriate coefficient, i.e.
15
T0 ~
1
h r0 / r00
Μ 0 m e r02 ,
(27)
where r00 is just a reference internuclear distance.
We have thence arrived to Eq.(19), we had previously obtained (References 8 and 9). Figures
1 – 7 are drawn based on this relationship. Note that the coefficient 4 2 is introduced on the
denominator of the RHS of Eq.(19) to install the similarity of this latter relationship with what
one would write on the basis of the Bohr Atom; here again, g is a coefficient insuring the
quality, which we expect to be the same for chemically alike molecules, which fortunately
happens indeed the case.
4. CONCLUSION
Herein we have proven a quantum mechanical theorem (stated at the beginning of the text),
on the basis of atomistic and molecular objects. We have used this theorem in our work
previously (cf. References 8, 9, 10, 18, 22 23), but it is here that we had, for first time the
opportunity of presenting a full scale proof of it. Thus, an arbitrary increase of masses in the
quantum description of an atomic or molecular object, leads to the increase of the related
total energy, and contraction of the size of the object at hand.
One may think that, the mass & energy equivalence of the STR, somewhat induces right away
the first part of our proof, i.e. an arbitrary mass increase would necessarily imply the increase
in the magnitude of the total energy, appearing as the eigenvalue of the quantum mechanical
description of concern, just as much. That is correct, but we have not crossed anywhere, any
quantum mechanical proof of it, similar to the one we have provided here. And the reason
seems to be simple. No one apparently, has considered to arbitrarily change masses that enter
in a quantum mechanical description. At the same time what is important to us, is that, our
result holds generally true, i.e. regardless whether the quantum mechanical description in
question, is relativistic or non-relativistic. And this does not seem trivial at all.
Quite on the contrary, it would not hold, if the description is relativistic, but the potential
energy terms input to it, are not compatible with the STR. In any case, the metric, due to the
mass change is altered. Lengths accordingly, are stretched. And this is something (to our
knowledge) totally novel.
Consequently we have proven that the occurrence we disclosed, yields the invariance of the
quantity [energy x mass x size2] (still, with respect to an arbitrary change in the quantum
mechanical description at hand). This is, though as well, a Lorentz invariant quantity,
whether mass that enters in it, is non-relativistic in the case of a non-relativistic description, or
relativistic in the case of a relativistic description.
This dual invariance of the quantity [energy x mass x size2] shapes a fundamental matter
architecture, again no matter what we deal with a relativistic or non-relativistic framework.
Thence we have the universal relationship E 0 M 0R 02 ~ h2 [Eq.(15)], where again, the clock
mass M0 is a rest mass in the case of a non-relativistic description, and is a relativistic mass in
the case of a relativistic description (cf. Table 1). Thereby our results can be easily extended
to the case of relativistic quantum mechanics. It can further be generalized to the case of
nuclear objects.
16
Note that, after all we have unveiled, Eq.(15) can even be directly written on simple
relativistic considerations. Indeed we can right away write that, in order to yield the end
results of the STR, energy, mass and size must already at rest, obey a given interrelation, i.e.
the product E 0 M 0R 02 must be nailed to a Lorentz invariant constant. This constant would be
just a number bearing the dimensions of the quantity E 0 M 0R 02 , if it were question of a wall
clock, a stick meter, and a weight we would have put next to each other, and bring them
altogether to a uniform translational motion, say along the direction of the stick meter. Then
indeed the product
[the energy of the wall clock] x [the mass of the weight] x [the length of the stick meter],
would be invariant no matter what the velocity of the motion they would be brought to, is.
But, what if these three quantities were already somehow, interrelated, just like they are in the
case, for instance, of the hydrogen atom, in fact any quantum mechanical entity? Then the
product E 0 M 0R 02 coming into play, would not just draw an arbitrary Lorentz invariant
quantity, but at the same time, a universal Lorentz invariant quantity, that takes place in the
description of the entity. Of course, for different quantum mechanical descriptions, different
universal Lorentz invariant quantities could have come into play. But nature does not seem to
be that versatile. Thence we land straight at Eq.(15).
By the same token we can affirm that, because matter delineates a given matter architecture,
i.e. that of Eq.(15), one should be able to tap the end results of the STR.
Thus briefly, the proof we presented herein, becomes important in many ways. First of all, it
appears to work as the mechanism of the end results of the STR, when the object initially
considered via a relativistic or non-relativistic quantum mechanical description, it does not
matter, is brought to a uniform translational motion. More essentially our Eq.(15), along with
the matter architecture it delineates, works as the basis of the Galilean Principle of Relativity
(cf. Reference 7).
Secondly, it seems to work as the mechanism of the end results of the GTR, were this object
embedded in a gravitational field. This fact alone, allowed us to develop a whole new theory
of gravitation, thus providing us with the possibility of describing both the atomistic world
and the celestial world, with the same mathematical tools, thus allowing the quantization of
gravitation (Reference 18). Note that, what we do here can be generalized to any field the
object at hand can interact with (References 22 and 23).
The figures we present below, based on Eq.(15), constitute, amongst other things, an example
to the matter architecture we disclosed. That is, clock energy, or similarly [cf. Eq.13)], clock
period of time, for any given entity’s internal dynamics, under consideration, must be
structured in just a given manner, with respect to the clock mass, carrying the clock labor, and
the size of clock space, this labor takes place in, for the given dynamics. Thus, these figures
show, how strikingly, the lowest classical vibrational period of time (T0) is linked to the clock
mass (M0me)(1/2) (composed of the product of, the nuclei reduced mass M0, by the electron
mass me), and the internuclear distance (r0), to provide us with the Lorentz invariant
interrelation [Eq.(27)] for all diatomic molecules (References 8 - 13), and delineating the
matter architecture we have disclosed. In this article, we have as well provided a quick and
new derivation of this specific interrelation [cf. Eqs. (20) – (27)], as an application of our
findings.
17
30
T0 x103c (cm)
25
Cs2
20
RbCs
Rb2
15
KRb
K2
10
NaK
Na2
5
LiNa
Li2
H2
0
0
20
40
60
1/2
(1/(r0/r00) )(M0
80
1/2
2
1/2
r0 ) (amu A2)
Figure 1 Period of alkali molecules versus
(1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of H2
18
5
T0 (x103c)
4
Te2
3
Se2
2
S2
SO
1
O2
0
0
10
20
30
40
(1/(r0/r00)1/2)(M01/2 r02) (amu1/2A2)
Figure 2 Period of (O2, S2, Se2, Te2) versus (1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of O2
19
1,4
T0 (x104c)
P2
1,2
1,0
0,8
PN
0,6
N2
0,4
0,2
0,0
0
2
4
6
8
10
12
(1/(r0/r00)1/2)(M01/2 r02) (amu1/2A2)
Figure 3
Period of (N2, PN, P2 ) versus (1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of N2
20
50
T0 (x104c)
I2
40
30
Br2
ICl
20
Cl2
BrF
F2
10
ClF
0
0
10
20
30
40
50
(1/(r0/r00)1/2)(M01/2 r02) (amu1/2A2)
Figure 4
Period of diatomic molecules, made of combinations of
halogen atoms, versus (1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of F2
21
60
T0 (x104c)
50
KBr
KI
40
KCl
RbCl
NaI
30
NaCl
NaBr
KF
20
10
0
0
10
20
30
40
1/2
(1/(r0/r00) )(M0
Figure 5
50
1/2
2
60
1/2
2
r0 ) (amu A )
Period of different alkali-halogen molecules
versus (1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of KF
22
80
T0 (x104c)
TlI
60
InI
TlBr
40
TlCl
InCl
AlBr
20
AlCl
BBr
BCl
BF
0
0
10
20
30
40
50
60
(1/(r0/r00)1/2)(M01/2 r02) (amu1/2A2)
Figure 6
Period of diatomic molecules, made of atoms belonging to
respectively the 3th and 7th columns of the periodic table,
versus (1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of BF
23
30
T0 (x104c)
25
SnS
PbS
20
PbO
15
SiS
SnO
10
GeO
SiO
CS
CO
5
0
0
5
10
15
20
25
(1/(r0/r00)1/2)(M01/2 r02) (amu1/2A2)
Figure 7
Period of diatomic molecules, made of atoms
belonging to respectively the 4th and 6th columns of the periodic table, versus
(1/(r0/r00)1/2)(M01/2 r02)
r00 : internuclear distance of CO
ACKNOWLEDGEMENT
The author would like to thank to Dear Friend Professor R. Tokay, who we lost, to Dr. Elman
Hasanov, Professor at Isik University, Istanbul, Turkey, to Dr. Alexander L Kholmetskii,
Professor at Belarus State University, Minsk, Belarus, to Dr. Metin Arik, Professor at
Bogazici University, Istanbul, Turkey, to Professor V. Rozanov, Director of the Laser Plasma
Theory Department, Lebedev Institute, Federation of Russia, to Professor N. Veziroglu,
Director of the Clean Energy Research Institute, University of Miami, to Dr. C. Marchal,
Research Director, ONERA, France, to Dr. O. Sinanoglu, Professor at Yale University, and to
Dr. S. Kocak, Professor at Anadolu University; without their unequal understanding, and
continuous support, this work would not have come to day light. Special thanks are further
due to Professor Fikret Aliev, Editor in Chief of the Applied and Computational Mathematics,
An International Journal, who cared to provide an fair review for this controversial article.
Thus I would like to extend as well, my gratitude to the referee Professor Aliev chose, to get
this article reviewed.
24
REFERENCES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
L. de Broglie, Recherches Sur La Théorie Des Quanta, Annales de Physique, 10e Série,
Tome III, 1925.
M. Planck, On the Law of Distribution of Energy in the Normal Spectrum, Annalen der
Physik, vol. 4, p. 553 ff, 1901.
A. Einstein, Title in English: On the Electrodynamics of Moving Bodies, Ann. Phys.,
17: 891, 1905.
J. C. Davis, Jr. “Advanced Physical Chemistry” (Chapter 10), The Ronald Press
Company, New York , 1965
R. M. Eisberg, Chaper 11, Fundamentals of Modern Physics, John Wiley & Sons, Inc.,
1961.
R. B. D. Evans, The Atomic Nucleus (Chapter 1), McGraw-Hill Book Company, 1955.
T. Yarman, Revealing The Mystery of The Galilean Principle Of Relativity, Part I: Basic
Assertions, International Journal of Theoretical Physics, Volume 48, Issue 8 (2009),
Page 2235.
T. Yarman, An Essential Approach to the Architecture of Diatomic Molecules. Part I.
Basic Theory, Optics and Spectroscopy, Volume 97 (5), 2004 (683).
T. Yarman, An Essential Approach To The Architecture of Diatomic Molecules, Part II:
How Size, Vibrational Period of Time And Mass Are Interrelated?, Optics and
Spectroscopy, Volume 97 (5), 2004 (691).
T. Yarman, Elucidation of the Complete Set of H2 Electronic States’ Ground Vibrational
Data, International Journal of Hydrogen Energy, 29, 2004 (1521).
T. Yarman, How Are Mass, Space Size, And Period of Time Structured, In Diatomic
Molecules?, Part I: Frame of The Approach, Balkan Physics Letters, 13-4, 2006.
T. Yarman, How Are Mass, Space Size, And Period of Time Structured, In Diatomic
Molecules?, Part II: Quantum Numbers of Electronic States, Balkan Physics Letters, 14-2,
2007.
T. Yarman, How Are Mass, Space Size, And Period of Time Structured, In Diatomic
Molecules?, Part III: A New Systematization, Balkan Physics Letters, 14-2, 2007.
G. Galileo, Dialogue Concerning the Two Chief World Systems – Ptlolemaic and
Copernican (Translated by S. Drake), University of California Press, Berkeley, 1953.
Isaak M. Yaglom, A simple Non-Euclidean Geometry and its Physical Basis: An
Elementary Account of Galilean Geometry and the Galilean Principle of Relativity, Abe
Shenitzer , New York: Springer-Verlag, ISBN 0387903321, 1979. (Translated from the
Russian Version.)
A. Einstein, Title in English: On the Electrodynamics of Moving Bodies, Ann. Phys.,
17: 891, 1905.
A. Einstein, The Meaning of Relativity, Princeton University Press, 1953.
T. Yarman, The End Results of the General Relativity Theory Via Just Energy
Conservation and Quantum Mechanics, Foundations of Physics Letters, December 2006.
G. Sobczyk, T. Yarman, Unification of Space-Time-Mass-Energy, Applied and
Computational Mathematics, An International Journal, December 2008.
A. Einstein, The Meaning of Relativity, Princeton University Press, 1953.
R. Mould, Basic Relativity (Chapter 12), Springer, 1994.
T. Yarman, A Novel Approach to The End Results of The General Theory of Relativity and
to Bound Muon Decay Retardation, DAMOP 2001 Meeting, Session J3, APS, May 16-19,
2001, London, Ontario, Canada
25
23
24
25
26
27
28
T. Yarman, A Novel Approach to the Bound Mon Decay Rate Retardation: Metric
Change Nearby the Nucleus, Physical Interpretations of the Theory of Relativity
Conference, Bauman Moscow State Technical University, 4 – 7 July 2005.
G. Herzberg, Molecular Spectra and Molecular Structure, Chapter 8, Volume I, D.Van
Nostrand Company, Inc, Reprint Edition, 1991.
Birge, Physic Rev. 25, 240 (1925).
R. Mecke, Z. Physics 32, 1925.
G. Herzberg, Molecular Spectra and Molecular Structure (D. Van Nostrand Company,
Inc., 1989), Vol. 1.
Born and Oppenheimer, Ann. Physik, 84, 457 (1927).
26